Olympiades ENSAE - 1er S 2025

  • Posted on: 8 May 2025
  • By: sbana

Exercice 1

Cet exercice est composé de parties $A$, $B$ et $C$  dans une large mesure indépendantes.
 
Partie A :

On définit par $A$ l'ensemble des fonctions $f\ :\  [0\;,1]\rightarrow \mathbb{R}$ vérifiant les conditions suivantes :

$\bullet\ $Pour tous réels $\alpha$ et $\beta\in[0\;,1]/\alpha <\beta$ si $f(\alpha)\times f(\beta)\leq 0$ alors il existe au moins $\overline{x}]\alpha\;,\beta[$ tel que $f\left(\overline{x}\right)=0$

$\bullet\ f(x)=f(1)=0$

$\bullet\ $Pour tout $x$ réel de l'intervalle $\left[0\;,\dfrac{7}{10}\right]\;,f\left(x+\dfrac{3}{10}\right)\neq f(x)$

Soit $f$ un élément de $A.$

On définit une fonction $h\ :\ \left[0\;,\dfrac{7}{10}\right]\rightarrow\;,\mathbb{R}$ donnée par : $h(x)=f\left(x+\dfrac{3}{10}\right)-f(x)$, pour tout $x\left[0\;,\dfrac{7}{10}\right]$

On suppose que h vérifie la condition $(1)$
 
$(1)$ Montrer que $h(x)$ est de signe constant sur $[0\;,\dfrac{7}{10}]$

$(2)$ Démontrer que l'équation $f(x)=0$ admet au moins sept solutions sur $[0\;,\dfrac{7}{10}]$
 
Partie B :

Les martiens sont les habitants, en nombre éventuellement infini, de la planète Mars.

Vis à vis d'eux-mêmes et de leurs semblables, les martiens sont capables de ressentir deux types d'émotions, qu'ils appellent amour et respect.

Il a été observé que :  

$\bullet\ $Chaque martien aime un et un seul martien, et respecte un et un seul martien.
 
$\bullet\ $Si $A$ aime $B$, alors tout martien qui respecte $A$ aime également $B$
 
$\bullet\ $ Si $A$ respecte $B$, alors tout martien qui aime $A$ respecte également $B.$
 
$\bullet\ $Chaque martien est aimé d'au moins un martien.  

On se propose de vérifier s'il est vrai que chaque martien respecte le martien qu'il aime.
 
Pour chaque martien $x$ , on désigne respectivement par $f(x)$ et $g(x)$ les martiens aimés et respectés par $x$

$(1)$ Montrer que les fonctions $f$ et $g$ sont bien définies de l'ensemble $X$ des martiens sur lui-même.
 
$(2)$ Montrer que $f[g(x)]=f(x)$ et $g[f(x)]=g(x)$ pour tout $x$ dans $X$
 
$(3)$ Montrer finalement que, pour tout $x$, on a $f(x)=g(x)=x$
 
$(4)$ Conclure !

Partie C :

Soit $n\in\mathbb{N}^{\ast}$, $m_{1}$, $m_{2}$, $\ldots$, $m_{n}$ des réels et $\mathrm{e}_{1}$, $\mathrm{e}_{2}$, $\ldots$, $\mathrm{e}_{n}$

des réels strictement, positifs on a :
$$\begin{array}{rcl}\dfrac{m_{1}^{2}}{\mathrm{e}_{1}}+\dfrac{m_{2}^{2}}{\mathrm{e}_{2}}+\ldots+\dfrac{m_{n}^{2}}{\mathrm{e}_{2}}\geq\dfrac{\left(m_{1}+m_{2}+\ldots+m_{n}\right)^{2}}{\mathrm{e}_{1}+\mathrm{2}+\ldots+\mathrm{e}_{n}}\end{array}$$

Cette inégalité est connue sous le nom de l'inégalité des Mauvais Élèves $(IME)$
En appliquant l'$IME$, montrer que :

$(1)$ Si $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ sont $n$ réels strictement positifs alors :

$\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\ldots+\dfrac{1}{a_{n}}\geq\dfrac{n^{2}}{a_{1}+a_{2}+\ldots+a_{n}}$
 
$(2)$ Si $a$, $b$ et $c$ sont $3$ réels strictement positifs alors :

$\dfrac{a}{a+2c}+\dfrac{b}{b+2a}+\dfrac{c}{c+2b}\geq 1$

$(3)$ Si $x$, $y$ et $z$ sont réels strictement positifs tels que : $\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}=1$ alors $xyz\geq 8$

On considère la figure suivante où $ABMN$ et $ARSC$ sont des carrés construits à partir du triangle $ABC$

On construit en outre le parallélogramme $ANA'R$

1. a. Démontrer que $\left(AA'\right)$ est une hauteur du triangle $ABC$

b. Démontrer que $AA'=BC$

2. On considère les parallélogramme $NARA'$, $QBMB'$ et $SCPC'$

Justifier que $\left(AA'\right)$, $\left(BB'\right)$ et $\left(CC'\right)$ sont concourantes.

b. Démontrer que dans un triangle $ABC$, si $M$ désigne le milieu de $[BC]$ alors : $AB^{2}+ac^{2}=2\left(BM^{2}+AM^{2}\right)$ (Théorème d'Apollonius).

c. En déduire que $NR^{2}+QM^{2}+SP^{2}=3\left(AB^{2}+AC^{2}+BC^{2}\right)$

Exercice 3

$\mathbb{R}$ désigne l'ensemble des nombres réels.

Dans ce problème, on cherche à déterminer les applications $f$ définies sur $]0\ ;\ +\infty[$ et à valeurs dans $]0\ ;\ +\infty[$  vérifiant les deux propriétés suivantes :

$\bullet\ $pour tous nombres réels strictement positifs $x$ et $y$, $f[xf(y)]=yf(x)$ ;
 
$\bullet\ f$ est bornée sur $[1\ ;\ +\infty[$ il existe un nombre réel $A$ tel que pour tout nombre réel $x\geq 1$, $f|f(x)|\leq A$

Partie I

Soit $I$ un intervalle de $\mathbb{R}$ et soit $h$ une application définie sur $I$ et à valeurs dans $I$

On dit que est $h$ une involution de $I$ si pour tout nombre réel $x$ dans $I\;,h[h(x)]=x$

1. Donner un exemple d'involution de $\mathbb{R}$ dans R autre que l'identité.

2. Donner un exemple d'involution de $]0\ ;\ +\infty[$ dans $]0\ ;\ +\infty[$ autre que l'identité.   

3. Montrer qu'une involution de $I$ dans $I$ est bijective.

Partie II

Soit $f$ une fonction vérifiant les deux conditions citées au début de l'énoncé.
 
1. Soit deux nombres réels $y_{1}$, $y_{2}$ strictement positifs tels que $f\left(y_{1}\right)=f\left(y_{2}\right).$

Montrer que $y_{1}f(1)=y_{2}f(1)$

2. Montrer que $f$ est injective.
 
3. Montrer que $f[f(1)]=f(1)$ puis $f(1)$ puis que $f(1)=1$
 
4. Montrer que $f$ est une involution de $]0\ ;\ +\infty[$

5. Soient $a$ et $b$ deux réels strictement positifs.

Montrer que $f(ab)=ff(a)f(b)$

Indication : on pourra poser $b=f(y)$

Partie III

On note $F$ l'ensemble des points fixes de $f$ : $F=\lbrace x\in]0\ ;\`+\infty\left[/f(x)=x\right\rbrace$
 
1) Montrer que pour tout $x\in]0\ ;\ +\infty[\;,xf(x)$ est un élément de  $F.$

2. Montrer que $1$ est un élément de $F.$
 
3. Montrer que si $x$ et $y$ sont des éléments de $F$, alors $xy$ et $\dfrac{x}{y}$
 sont également des éléments de $F.$
 
4. Montrer que si $x$ est un élément de $F$, alors pour tout entier naturel $n$, $xn$ est un élément de $F$

5. Montrer que si $x$ est un élément de $F$, alors $x=1$

Indication : on pourra considérer l'application : $\mathbb{N}\rightarrow \mathbb{N}$ définie par $x_{n}x^{n}$

6. Montrer que $F=\lbrace 1\rbrace$
 
7. En déduire $f.$
 
8. Donner enfin toutes les applications répondant au problème posé.

 

 

Niveau: 
Classe: 
Type: 
Série: 

Commentaires

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“The Russian war machine continues to destroy lives despite everything,” Zelensky said in a statement, hours before he’s due to meet US President Donald Trump in the Oval Office. “That is precisely why we are seeking assistance to put an end to the killings. That is why reliable security guarantees are required. That is why Russia should not be rewarded for its participation in this war.”
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“Everyone seeks dignified peace and true security,” the Ukrainian president said. “And at this very moment, the Russians are attacking Kharkiv, Zaporizhzhia, the Sumy region, and Odesa, destroying residential buildings and our civilian infrastructure.”

At least seven people were killed in Russia’s attack? on Kharkiv and a further three killed in the ballistic missile strike on the city of Zaporizhzhia, with scores more injured, according to Ukrainian authorities.

“This was a demonstrative and cynical Russian strike,” Zelensky added.
kra25 at
https://kra25-c.cc

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“Everyone seeks dignified peace and true security,” the Ukrainian president said. “And at this very moment, the Russians are attacking Kharkiv, Zaporizhzhia, the Sumy region, and Odesa, destroying residential buildings and our civilian infrastructure.”

At least seven people were killed in Russia’s attack? on Kharkiv and a further three killed in the ballistic missile strike on the city of Zaporizhzhia, with scores more injured, according to Ukrainian authorities.

“This was a demonstrative and cynical Russian strike,” Zelensky added.
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https://kra28at.net

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<h3>Transaxle vs. Transmission and Differential</h3>

Да, действительно. Так бывает. Можем пообщаться на эту тему. Здесь или в PM.
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While many visitors come to the area for the otherworldly landscapes of Badlands National Park, roughly 60 miles east of the city, or the wildlife viewing at Custer State Park to the south, the city offers its own unique scenery.

Rapid City borders the Black Hills to the west and prairie grasslands to the east. Rapid Creek meanders through town and an adjacent greenway connects much of the city’s 1,650 acres of park land.
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Two beloved decades-old parks — Dinosaur Park and Storybook Island — have recently been upgraded to be disability accessible. Wheelchair-friendly Dinosaur Park includes seven life-size dinosaur statues and panoramic city views. At Storybook Island, six pieces of playground equipment are wheelchair accessible and have Braille panels and sensory panels for children with autism. The park is renowned for fairy tale character playsets, an antique carousel and a miniature train. Admission to both parks is free.
https://tripskan40.cc
трипскан
Mountain bikers and hikers favor the panoramic views from trails within the 150-acre Skyline Wilderness Area, or from spots along the more than 20 miles of trails in Hanson-Larsen Memorial Park, where M Hill pays tribute to science and engineering university South Dakota Mines. All of these trails are located within city limits.

Jason Kingsbury is a tourist-turned-resident who relocated largely because of the city’s outdoor recreation. Kingsbury is an avid camper, mountain biker and fly fisherman.

“I was absolutely blown away with how much there is to do in Rapid City,” Kingsbury said. Accessing outstanding outdoor recreation in just minutes is unique, he said.

“A lot of people do not have that experience. They can go to Memorial Park and ride world-class (trails). They can ride there from their hotel — that really impresses people,” Kingsbury said.

“What a lot of people always say is ‘I can’t believe how cool this is. I never thought South Dakota had things like this,’” he said. “They realize real quickly we’re far more than just Mount Rushmore.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
трипскан
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>tripscan</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
трипскан
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан сайт</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
tripscan top
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>tripscan top</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
tripscan top
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>tripscan top</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
tripskan
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

Найти надежные грузоперевозки важно для бизнеса и частных перевозок. В этой статье — практические советы, проверенные методы и критерии выбора.

Определите требования к перевозке
Сначала уточните параметры груза: вес, габариты, ценность, требуемые сроки и условия (температура, хрупкость). Чем точнее описание, тем проще выбрать перевозчика.

Поиск исполнителей
Используйте несколько каналов:
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Оцените репутацию и документы
Проверьте у перевозчика:
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- Страхование ответственности.
- Отзывы клиентов и портфолио задач похожего масштаба.
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Заключайте письменный договор, где прописаны сроки, маршрут, условия перевозки, ответственность сторон и порядок урегулирования споров. Требуйте товарно-транспортные документы и акты приема-передачи <a href=http://perevezem-bai.belorussia.su/>http://perevezem-bai.belorussia.su/</a>

Контроль и коммуникация
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Советы по снижению рисков
- Делайте предварительные тестовые перевозки.
- Выбирайте компании с опытом в вашем сегменте.
- Страхуйте ценное имущество.
- Держите запас времени на случай задержек.

Выбор надежных грузоперевозок требует тщательной проверки, сравнения предложений и ясных договорных условий. Следуя этим шагам, вы минимизируете риски и получите стабильный результат.

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Выберите способ, удобный вам: платёжная система, а также вывод на карту.

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Совет
Выбирайте надёжные криптообменники и всегда проверяйте детали транзакций перед подтверждением.

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
трипскан сайт
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан вход</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
трипскан сайт
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

While many visitors come to the area for the otherworldly landscapes of Badlands National Park, roughly 60 miles east of the city, or the wildlife viewing at Custer State Park to the south, the city offers its own unique scenery.

Rapid City borders the Black Hills to the west and prairie grasslands to the east. Rapid Creek meanders through town and an adjacent greenway connects much of the city’s 1,650 acres of park land.
<a href=https://tripskan40.cc>трип скан</a>
Two beloved decades-old parks — Dinosaur Park and Storybook Island — have recently been upgraded to be disability accessible. Wheelchair-friendly Dinosaur Park includes seven life-size dinosaur statues and panoramic city views. At Storybook Island, six pieces of playground equipment are wheelchair accessible and have Braille panels and sensory panels for children with autism. The park is renowned for fairy tale character playsets, an antique carousel and a miniature train. Admission to both parks is free.
https://tripskan40.cc
трипскан вход
Mountain bikers and hikers favor the panoramic views from trails within the 150-acre Skyline Wilderness Area, or from spots along the more than 20 miles of trails in Hanson-Larsen Memorial Park, where M Hill pays tribute to science and engineering university South Dakota Mines. All of these trails are located within city limits.

Jason Kingsbury is a tourist-turned-resident who relocated largely because of the city’s outdoor recreation. Kingsbury is an avid camper, mountain biker and fly fisherman.

“I was absolutely blown away with how much there is to do in Rapid City,” Kingsbury said. Accessing outstanding outdoor recreation in just minutes is unique, he said.

“A lot of people do not have that experience. They can go to Memorial Park and ride world-class (trails). They can ride there from their hotel — that really impresses people,” Kingsbury said.

“What a lot of people always say is ‘I can’t believe how cool this is. I never thought South Dakota had things like this,’” he said. “They realize real quickly we’re far more than just Mount Rushmore.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
трипскан сайт
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

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For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>tripscan</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
tripscan
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

For people who cannot or choose not to breastfeed, infant formula is lifesaving, supplying their babies with precious nutrients they need to grow and thrive.

Concerned about contaminants and the use of controversial seed oils in the nation’s tightly regulated infant formula supply, US Health and Human Services Secretary Robert F. Kennedy Jr. has launched a review of infant formula regulations he calls Operation Stork Speed.
<a href=https://trip-scan39.cc>трипскан</a>
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
https://trip-scan39.cc
tripscan
Yet while industry and experts debate those regulations, there is another type of “formula” — for toddlers ages 1 to 3 years old — that no one is overseeing.

“Toddler milks” are marketed to parents who have been using infant formula as a necessary next step in their child’s nutritional journey, according to the American Academy of Pediatrics. Critics say these expensive concoctions — first introduced into the United States in the 1990s — are not nutritionally necessary, may be contributing to childhood obesity, and should not be sold at all.
In fact, some toddler beverages may be potentially harmful to young children due to excess added sugars, defined as sweeteners that don’t naturally occur in food, Fuchs said.

The recommended amount of added sugar for infants and children younger than age 2 is zero, according to the US Centers for Disease Control and Prevention. Yet research from 2019 found that on average, infants consume 1 teaspoon of added sugar a day while toddlers consume about 6 teaspoons a day.

“Infants and children exposed to high sugar, fat or salt in their diets develop a preference for these as they grow older,” Fuchs said. “Studies show this is a risk for childhood obesity and all of its consequences through adulthood.”

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>tripskan</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
трипскан сайт
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>tripskan</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
tripscan
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>трипскан сайт</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
tripskan
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

<h3>Why is Diameter Important?</h3>

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</div>

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>трип скан</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
трипскан вход
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>трипскан вход</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
tripscan top
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>трипскан сайт</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
трипскан
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

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Cracker Barrel’s modern makeover doesn’t stop with redoing its restaurants. It’s dropping the barrel and the man from its logo, too.

On Tuesday, the Southern-inspired casual dining chain unveiled a new logo “rooted even more closely to the iconic barrel shape,” but without the barrel itself — a central part of the brand’s identity since 1977. (As for the the barrel itself, it was “essentially the water coolers of the day,” Cracker Barrel explained in a blog post.)
<a href=https://tripskan39.cc>tripskan</a>
The identity refresh also includes new TV commercials, a redesigned menu and several new fall-themed foods, part of a larger $700 million transformation plan to shake off its stodgy image and lure in new diners.

“The way we communicate, the things on the menu, the way the stores look and feel … all of these things came up time and time again in our research as opportunities for us to really regain relevancy,” said CEO Julie Felss Masino in 2024.
https://tripskan39.cc
трипскан сайт
In particular, the new logo is the latest in a string of changes angering some of its loyal fans who fear the 56-year-old chain is drifting too far from its bucolic roots. On social media, some users griped, with one writing that the “changing the logo just feels like another little piece of culture dying off.” The change also angered some conservatives, too, like President Donald Trump’s son.

Cracker Barrel has also been remodeling some of its 660-plus restaurants. The chain has “decluttered” the interiors by removing the country-themed trinkets that lined the walls and lightened up the interiors, shifting away from the dark woods. So far, reaction has been mixed on social media, with some videos on TikTok going viral voicing their displeasure.

Masino remains adamant that the renovations are working, telling ABC News this week that “people like what we’re doing” and that feedback has been “overwhelmingly positive.”

In June, Cracker Barrel posted an unusual earnings report for a restaurant: It’s taking a $5 million hit from tariffs because of its retail shops, which largely has products imported from overseas. Restaurant revenue and same-store sales both slightly grew, mirroring other increases casual dining chains are experiencing.

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Orlando officials condemn removal of rainbow crosswalk near Pulse nightclub mass shooting site as a ‘cruel political act’
<a href=https://mista.ua/%D0%98%D0%BD%D1%84%D1%80%D0%B0%D1%81%D1%82%D1%80%D1%83%...Кредитный адвокат Запорожье</a>
Orlando officials on Thursday condemned the state’s overnight removal of a rainbow crosswalk outside the Pulse nightclub – a commemoration of the 49 people killed at the LGBTQ-friendly nightclub in 2016.

Painting over one of the most important landmarks of the state’s LGBTQ community is part of a larger attack on LGBTQ people by Gov. Ron DeSantis and his administration, officials say.
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Orlando Mayor Buddy Dyer said he was “devastated” by the removal of the Pulse memorial crosswalk.

“This callous action of hastily removing part of a memorial to what was at the time our nation’s largest mass shooting, without any supporting safety or discussion, is a cruel political act,” Dyer said in a post on X.

The crosswalk was installed by the state in 2017 and adheres to national safety standards, Dyer added. The colorful crosswalk added visibility, making it safer for the large number of visitors to the memorial, he said.

CNN has reached out to the Florida Department of Transportation for comment.

Related article
rainbow pulse memorial
A rainbow rose over the Pulse nightclub memorial on the fifth anniversary of the mass shooting at the Orlando venue

The removal comes only two months after the city marked the nine-year anniversary of the Pulse tragedy. On June 12, 2016, a gunman opened fire at the popular gay nightclub, making it the worst mass shooting in modern US history at the time.

In a photo from 2021, the crosswalk painted in Pride colors in front of the of the onePULSE Foundation's Pulse Interim Memorial, honoring victims of the nightclub shooting.
In a photo from 2021, the crosswalk painted in Pride colors in front of the of the onePULSE Foundation's Pulse Interim Memorial, honoring victims of the nightclub shooting. Phelan M. Ebenhack/AP/File
The removal also comes during campaigns by the state and Trump administration to paint over “asphalt art,” including rainbow crosswalks. The dual directives are calling this a safety measure to make roads easier to navigate without distractions.

In June, the Florida Department of Transportation issued a memo prohibiting crosswalk markings and pavement surface art “associated with social, political, or ideological messages or images and does not serve the purpose of traffic control.”

The following month, US Transportation Secretary Sean Duffy sent a letter to governors of all 50 states saying intersections and crosswalks should be “kept free from distractions.”

“Taxpayers expect their dollars to fund safe streets, not rainbow crosswalks,” Duffy said about the letter in an X post on July 1. “Political banners have no place on public roads. I’m reminding recipients of USDOT roadway funding that it’s limited to features advancing safety, and nothing else.”

<h2>Potential Causes of Transmission Issues</h2>

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