Olympiades ENSAE - 1er S 2025
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.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
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That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
“You’ve got to do it for prevention,” she said. “You have to learn something from this.”
That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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Preventing murder starts with understanding killers. These women
A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
“You’ve got to do it for prevention,” she said. “You have to learn something from this.”
That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
“You’ve got to do it for prevention,” she said. “You have to learn something from this.”
That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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Preventing murder starts with understanding killers. These women
A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
“You’ve got to do it for prevention,” she said. “You have to learn something from this.”
That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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Preventing murder starts with understanding killers. These women
A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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A month after Lyle and Erik Menendez were arrested for brutally slaying their parents inside their Beverly Hills home, Dr. Ann Burgess entered the Los Angeles County Jail with a stack of blank paper and a set of colored pencils.
It was April 1990, and the maelstrom around Jose and Kitty Menendez’s double murder – and the brothers’ forthcoming trial – had reached a fever pitch. News articles described the crime scene in gory, painstaking detail. Prosecutors and tabloids portrayed the brothers as greedy, calculated, cold-blooded killers.
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Burgess was among the earliest women to work with the FBI and a key member of what was known as the bureau’s Behavioral Science Unit in the late ’70s.
That team has since been dubbed “Mindhunters” because they willingly delve into the darkest parts of the human psyche to better understand what motivates a murderer. What they uncover could make even the most hardened detectives blanch.
And while criminal profiling is not an exact science, it is a method investigators increasingly lean on to identify warning signs of a would-be killer.
CNN spoke to former profilers – all women like Dr. Burgess who worked with the FBI – who have pioneered and practiced ways to connect the dots between evidence and psychology to help solve and prevent crimes.
“You start very slowly,” the now 88-year-old told CNN of her approach with Menendez. “You start with, ‘How far back can you remember?’ … and gradually get up to, ‘When did you first have this idea of what you wanted to do to your parents?’”
Burgess said she spent 50 hours interviewing Menendez and, as she recounts in her latest book, she was later called as an “expert witness” to testify about how Erik and Lyle’s decision to confront their father over what they alleged was years of sexual abuse could have provoked enough fear for them to commit a double murder.
She’s since been accused of profiling Menendez as a way to excuse or justify the brothers’ crimes, but Burgess staunchly rejects that characterization.
“You’ve got to do it for prevention,” she said. “You have to learn something from this.”
That, she says, is the question that drives most criminal profilers: How can we prevent the next murder?
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