LFM Pure

grandes-ecoles 2016 QIII.C.4 Proof of Equivalence or Logical Relationship Between Conditions View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that $P \in \mathbb{R}_n[X]$ is integer-valued on the integers if and only if its coordinates in the basis $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ are integers.

grandes-ecoles 2016 QIII.C.5 Deduction or Consequence from Prior Results View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $P \in \mathbb{R}[X]$ of degree $d \in \mathbb{N}$. Show that if $P$ is integer-valued on the integers then $d! P$ is a polynomial with integer coefficients. Study the converse.

grandes-ecoles 2016 QIV.B.1 Deduction or Consequence from Prior Results View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that for every integer $k$ strictly positive, $k^\alpha$ belongs to $\mathbb{N}^*$.

grandes-ecoles 2016 QIV.B.2 Direct Proof of an Inequality View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that $\alpha$ is positive or zero.

grandes-ecoles 2016 QIV.B.3 Higher-order or nth derivative computation View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.

grandes-ecoles 2016 QIV.C.1 Proof Involving Combinatorial or Number-Theoretic Structure View

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$ (where $\lfloor \cdot \rfloor$ denotes the floor function). We now choose $x \in \mathbb{N}^*$.
Show that the expression $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f_\alpha(x+j)$$ is a relative integer.

grandes-ecoles 2016 QIV.C.3 Deduction or Consequence from Prior Results View

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$.
Conclude that $\alpha$ is a natural number.

grandes-ecoles 2016 Q6b Convergence of Expectations or Moments View

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The function $f$ is the pointwise limit of the sequence $f_n$. Show that the function $f$ satisfies the following equality on $\mathbb{R}$ $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

grandes-ecoles 2016 Q7a Expectation of a Function of a Discrete Random Variable View

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$. Show that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, we have $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$.

grandes-ecoles 2016 Q7b Probability Bounds and Inequalities for Discrete Variables View

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$, and such that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Deduce that if moreover the support of $h$ is included in $\mathbb{R}^+$, then for all $x \in \mathbb{R}$, $h(x) = 0$.

grandes-ecoles 2016 Q7c Existence of Expectation or Moments View

Conclude that there exists a unique bounded function with support in $\mathbb{R}^+$ solution of $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$

grandes-ecoles 2016 Q8 Existence Proof View

Let $V \geqslant 0$ be a real number.
8a. Give an example of an integer simplex in $\mathbb{R}^2$ with volume greater than or equal to $V$ and having no interior integer points.
8b. Give an example of an integer simplex in $\mathbb{R}^3$ with volume greater than or equal to $V$ whose only integer points are the vertices.

grandes-ecoles 2016 Q8a Existence of Expectation or Moments View

Show that the set $\Lambda_X := \bigcup_{n \in \mathbb{N}} \left\{y \in \mathbb{R} \mid \mathbb{P}\left(S_n = y\right) > 0\right\}$ is countable and included in $\mathbb{R}^+$.

grandes-ecoles 2016 Q8b Expectation of a Function of a Discrete Random Variable View

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$

grandes-ecoles 2016 Q9 Proof of Set Membership, Containment, or Structural Property View

Let $\mathcal{K}$ be a compact convex set in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{K}}$.
9a. Show that the set of $\lambda \geqslant 0$ such that $-\lambda \mathcal{K} \subset \mathcal{K}$ is an interval.
We denote $$a(\mathcal{K}) = \sup\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$$
9b. Show that $a(\mathcal{K}) < \infty$ and that $a(\mathcal{K}) = \max\{\lambda \geqslant 0 \mid -\lambda \mathcal{K} \subset \mathcal{K}\}$.
9c. Show that $0 < a(\mathcal{K}) \leqslant 1$. Deduce that $a(\mathcal{K}) = 1$ if and only if $\mathcal{K}$ is symmetric with respect to 0.

grandes-ecoles 2016 Q10a Direct Proof of a Stated Identity or Equality View

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ such that $$\forall (x, y) \in \Lambda^2, \quad x + y \in \Lambda$$ We say that $\Lambda$ is closed under addition. Show that if $(x, y) \in \Lambda^2$, $(k, n) \in \mathbb{N} \times \mathbb{N}^*$ and $k \leqslant n$, then $nx + k(y-x) \in \Lambda$.

grandes-ecoles 2016 Q11 Proof of Equivalence or Logical Relationship Between Conditions View

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that two integer simplexes $\mathcal{S}$ and $\mathcal{S}'$ are equivalent if and only if there exist a matrix $A \in \mathrm{GL}_n(\mathbb{Z})$ and a vector $b \in \mathbb{Z}^n$ such that $\mathcal{S}' = A(\mathcal{S}) - b$.

grandes-ecoles 2016 Q11 Matrix Norm, Convergence, and Inequality View

We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Show that $| \alpha | \leq 1$ with equality if and only if $q = r = 0$.

grandes-ecoles 2016 Q12 Proof That a Map Has a Specific Property View

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that the volume, the number of integer points and the number of interior integer points are the same for two equivalent integer simplexes.

grandes-ecoles 2016 Q12 Prove a general algebraic or analytic statement by induction View

We prove Broyden's theorem by induction on the dimension. We assume the result holds up to rank $n - 1$ and we write $O$ in the form of a block matrix $$O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$$ where $P \in M _ { n - 1 } ( \mathbb { R } )$ and thus $r , q \in \mathbb { R } ^ { n - 1 }$ and $\alpha \in \mathbb { R }$.
Treat the case $| \alpha | = 1$.

grandes-ecoles 2016 Q12a Existence Proof View

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$

grandes-ecoles 2016 Q12b Proof That a Map Has a Specific Property View

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Suppose that for every sequence $\left(x_n\right)_{n \geqslant 0}$ with values in $\Lambda$ such that $x_n \rightarrow +\infty$, $f\left(x_n\right) \rightarrow 0$ when $n \rightarrow +\infty$. Show that $f(x) \rightarrow 0$ when $x \rightarrow +\infty$.

grandes-ecoles 2016 Q13 Matrix Algebra and Product Properties View

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that ${ } ^ { t } P P + q { } ^ { t } q = I _ { n - 1 }$, ${ } ^ { t } P r + \alpha q = 0$ and ${ } ^ { t } r r + \alpha ^ { 2 } = 1$.

grandes-ecoles 2016 Q13a Proof of Distributional Properties or Symmetry View

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. We consider a function $h$ uniformly continuous and bounded on $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $h(x) \leqslant h(0)$ and $$h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$$ We recall that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Show that for all $n \in \mathbb{N}$ and $x \geqslant 0$ such that $\mathbb{P}\left(S_n = x\right) > 0$, we have $h(-x) = h(0)$.

grandes-ecoles 2016 Q15 Matrix Algebra and Product Properties View

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that $${ } ^ { t } Q _ { + } Q _ { - } = I _ { n - 1 } - \frac { 2 } { 1 - \alpha ^ { 2 } } q { } ^ { t } q$$ and deduce that $$Q _ { - } = Q _ { + } - \frac { 2 } { 1 - \alpha ^ { 2 } } Q _ { + } q { } ^ { t } q$$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164