LFM Pure

grandes-ecoles 2022 Q1c Proof That a Map Has a Specific Property View

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are periodic with period 1.

grandes-ecoles 2022 Q1d Proof That a Map Has a Specific Property View

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.

grandes-ecoles 2022 Q1.1 Existence Proof View

Let $a$ be a real number in the open interval $]0,1[$. Show that there exists $\lambda > 0$ such that the polynomial $$P(x) = x - \lambda x(x-a)(x-1)$$ satisfies the following two properties:

$P([0,1]) = [0,1]$,
$P$ is increasing on $[0,1]$.

grandes-ecoles 2022 Q1.2 Computation of a Limit, Value, or Explicit Formula View

We fix a choice of $\lambda$ such that $P_a(x) = x - \lambda x(x-a)(x-1)$ satisfies $P([0,1])=[0,1]$ and $P$ is increasing on $[0,1]$. Let $\left(P_a^{\circ n}\right)_{n\geq 0}$ be the sequence of polynomials defined recursively by $P_a^{\circ 0}(x) = x$ and $P_a^{\circ n+1}(x) = P_a\left(P_a^{\circ n}(x)\right)$.
Show that $P_a^{\circ n}$ converges uniformly to 1 on every compact subset of $]a,1]$ and uniformly to 0 on every compact subset of $[0,a[$.

grandes-ecoles 2022 Q1.3 Proof of Set Membership, Containment, or Structural Property View

We denote by $\mathcal{C}([-1,1])$ the vector space of continuous functions from $[-1,1]$ to $\mathbb{C}$ and $\mathcal{T}([-1,1])$ the complex vector subspace of $\mathcal{C}([-1,1])$ generated by the functions $$e_k : t \mapsto e^{i\pi k t}, \quad k \in \mathbb{Z}.$$ Show that $\mathcal{T}([0,1])$ is a subalgebra of $\mathcal{C}([-1,1])$ for the usual multiplication law of functions.

grandes-ecoles 2022 Q1.4 Deduction or Consequence from Prior Results View

Let $b \in \mathbb{R}$ such that $\cos(b) \in ]0,1[$. Show that the sequence of functions $(f_{b,n})_{n\in\mathbb{N}}$ defined by $$f_{b,n}(t) = P_{\cos(b)}^{\circ n}\left(\cos^2\left(\frac{\pi}{2}t\right)\right)$$ converges uniformly to 1 on every compact subset of $]-\cos(b), \cos(b)[$ and converges uniformly to 0 on every compact subset of $[-1,-\cos(b)[\cup]\cos(b),1]$.

grandes-ecoles 2022 Q1.5 Existence Proof View

We denote by $\mathcal{C}([-1,1]^2)$ the space of continuous functions from $[-1,1]^2$ to $\mathbb{C}$ and $\mathcal{T}([-1,1]^2)$ the subspace generated by the functions $$e_{u,v} : (s,t) \mapsto e^{i\pi us}e^{i\pi vt}, \quad (u,v) \in \mathbb{Z}^2.$$ Let $a,b,c,d \in [-1,1]$ such that $a < b$ and $c < d$. Show that for every $\varepsilon < \min\left(\frac{b-a}{2}, \frac{d-c}{2}\right)$, there exists $f_\varepsilon \in \mathcal{T}([-1,1]\times[-1,1])$ satisfying the following properties:

$f_\varepsilon(s,t) \in [0,1]$ for all $(s,t) \in [-1,1]^2$,
$f_\varepsilon(s,t) \leq \varepsilon$ for $(s,t) \notin [a,b]\times[c,d]$,
$f_\varepsilon(s,t) \geq 1-\varepsilon$ for $(s,t) \in [a+\varepsilon,b-\varepsilon]\times[c+\varepsilon,d-\varepsilon]$.

grandes-ecoles 2022 Q2 Direct Proof of a Stated Identity or Equality View

Show that, for all $i$ and $k$ in $\llbracket 1 , n \rrbracket$, $$L _ { i } \left( a _ { k } \right) = \begin{cases} 1 & \text { if } k = i \\ 0 & \text { otherwise } \end{cases}$$ where $L_i(X) = \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \frac { X - a _ { j } } { a _ { i } - a _ { j } }$.

grandes-ecoles 2022 Q2 Proof of Equivalence or Logical Relationship Between Conditions View

Let $P$ be a polynomial function not identically zero with real coefficients. Show that the restriction of $P$ to $\mathbb { R } _ { + } ^ { * }$ belongs to $E$ if and only if $P ( 0 ) = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

grandes-ecoles 2022 Q3 Direct Proof of an Inequality View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have $$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2$$ and deduce that $\operatorname{proj}_C$ is continuous.

grandes-ecoles 2022 Q3 Direct Proof of an Inequality View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Show that for all $(x_1, x_2) \in \mathbb{R}^d \times \mathbb{R}^d$, we have $$\left(\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right) \cdot \left(x_1 - x_2\right) \geqslant \left\|\operatorname{proj}_C(x_1) - \operatorname{proj}_C(x_2)\right\|^2,$$ and deduce that $\operatorname{proj}_C$ is continuous.

grandes-ecoles 2022 Q3 Deduction or Consequence from Prior Results View

Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.

grandes-ecoles 2022 Q3 Direct Proof of a Stated Identity or Equality View

Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.

grandes-ecoles 2022 Q3 Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. For all $x \in [0,1]$ and $f \in \mathcal{C}^{2}([0,1])$, prove the inequality $$\left|f^{\prime}(x) - \frac{f\left(x_{2}\right) - f\left(x_{1}\right)}{x_{2} - x_{1}}\right| \leqslant \left\|f^{\prime\prime}\right\|_{\infty}.$$

grandes-ecoles 2022 Q3 Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. For all $x \in [0,1]$ and $f \in \mathcal{C}^2([0,1])$, prove the inequality $$\left|f^\prime(x) - \frac{f\left(x_2\right) - f\left(x_1\right)}{x_2 - x_1}\right| \leqslant \left\|f^{\prime\prime}\right\|_\infty.$$

grandes-ecoles 2022 Q3 Direct Proof of a Stated Identity or Equality View

Show that, for all $i \in \llbracket 1 , n \rrbracket$ and all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$\left\langle L _ { i } , P \right\rangle = P \left( a _ { i } \right).$$

grandes-ecoles 2022 Q3 Proof of Equivalence or Logical Relationship Between Conditions View

Let $a$ and $b$ be two real numbers. Show that the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & a \mathrm { e } ^ { t } + b \end{array} \right.$ belongs to $E$ if and only if $a = b = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

grandes-ecoles 2022 Q3a Existence Proof View

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.

grandes-ecoles 2022 Q3b Existence Proof View

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M.$$

grandes-ecoles 2022 Q3a Existence Proof View

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the function $D$ extends by continuity to a function $\widetilde{D}$ on $\mathbb{R}$ such that $\widetilde{D}(0) = 0$.

grandes-ecoles 2022 Q3b Existence Proof View

Let $D = f - g$ where $f(x) = \pi \operatorname{cotan}(\pi x)$ and $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and let $\widetilde{D}$ be its continuous extension to $\mathbb{R}$. Justify the existence of $\alpha \in [0,1]$ such that $\widetilde{D}(\alpha) = M$, where $M = \sup_{t \in [0,1]} \widetilde{D}(t)$, then show that: $$\forall n \in \mathbb{N}, \quad \widetilde{D}\left(\frac{\alpha}{2^n}\right) = M$$

grandes-ecoles 2022 Q4 Direct Proof of a Stated Identity or Equality View

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?

grandes-ecoles 2022 Q4 Direct Proof of a Stated Identity or Equality View

We fix two orthonormal families $u = (u_1, \ldots, u_p)$ of vectors of $V$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ satisfying the conditions of question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\det(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\det(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?

grandes-ecoles 2022 Q4 Computation of a Limit, Value, or Explicit Formula View

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Explicitly determine $\operatorname{proj}_C$ in the following cases: $$\text{i) } C = \mathbb{R}_+^d, \quad \text{ii) } C = \left\{y \in \mathbb{R}^d : \|y\| \leqslant 1\right\}$$ $$\text{iii) } C = \left\{y \in \mathbb{R}^d : \sum_{i=1}^d y_i \leqslant 1\right\}, \quad \text{iv) } C = [-1,1]^d$$

grandes-ecoles 2022 Q5 Proof of Set Membership, Containment, or Structural Property View

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that $D - C$ is a convex closed subset of $\mathbb{R}^d$ not containing $0$.

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

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Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

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Chain Rule 113

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Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144