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2025
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2024
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2023
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2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
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2020
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2019
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2018
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2017
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2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2025 centrale-maths1__official

40 maths questions

Q1 Number Theory Prime Counting and Distribution View
Let $n \in \mathbb { N } ^ { * }$. Show that
$$\prod _ { \substack { n + 2 \leqslant p \leqslant 2 n + 1 \\ p \text { prime } } } p \leqslant \binom { 2 n + 1 } { n } \leqslant 4 ^ { n } .$$
Q2 Number Theory Prime Counting and Distribution View
Show that, for all $n \in \mathbb { N } ^ { * }$,
$$\prod _ { \substack { p \leqslant n \\ p \text { prime } } } p < 4 ^ { n } .$$
One may proceed by induction and perform the inductive step by discussing according to the parity of $n$.
Q3 Number Theory Prime Counting and Distribution View
Deduce that, for all real $x \geqslant 1$,
$$\prod _ { \substack { p \leqslant x \\ p \text { prime } } } p < 4 ^ { x } .$$
Q4 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View
Show that, for all $n \in \mathbb { N } ^ { * }$,
$$\frac { 4 ^ { n } } { 2 n } \leqslant \binom { 2 n } { n } < 4 ^ { n } .$$
Q5 Number Theory Divisibility and Divisor Analysis View
Let $p$ be a prime number. Show that, for all $n \in \mathbb { N }$,
$$v _ { p } ( n ! ) = \sum _ { k = 1 } ^ { + \infty } \left\lfloor \frac { n } { p ^ { k } } \right\rfloor$$
Q6 Number Theory Divisibility and Divisor Analysis View
Deduce that, for all $n \in \mathbb { N } ^ { * } , k \in \mathbb { N }$ and $p$ prime number: if $p ^ { k }$ divides $\binom { 2 n } { n }$, then $p ^ { k } \leqslant 2 n$.
Q7 Number Theory Prime Counting and Distribution View
Let $n \in \mathbb { N } ^ { * }$. Justify that
$$\prod _ { \substack { p \leqslant n \\ p \text { prime } } } p \geqslant \prod _ { \substack { \sqrt { n } < p \leqslant n \\ p \text { prime } } } p$$
Q8 Number Theory Prime Counting and Distribution View
Deduce that, for all $n \in \mathbb { N } ^ { * }$,
$$n ^ { ( \pi ( n ) - \pi ( \sqrt { n } ) ) / 2 } < 4 ^ { n }$$
Q9 Number Theory Prime Counting and Distribution View
Let $n \in \mathbb { N } , n \geqslant 2$. Justify that
$$\pi ( \sqrt { n } ) \leqslant \sqrt { n } < \frac { n } { \ln ( n ) }$$
then deduce that
$$\pi ( n ) \leqslant 4 \frac { \ln ( n ) } { n }$$
One may note that $2 > \ln ( 4 )$.
Q10 Number Theory Prime Counting and Distribution View
Let $x \geqslant 3$. Using the monotonicity of the function $t \mapsto \frac { t } { \ln ( t ) }$ on the interval $[ \mathrm { e } , + \infty [$, show that
$$\pi ( x ) \leqslant 4 \frac { x } { \ln ( x ) }$$
Q11 Number Theory Prime Counting and Distribution View
Let $n \in \mathbb { N } ^ { * }$. Show that
$$\binom { 2 n } { n } \leqslant ( 2 n ) ^ { \pi ( 2 n ) }$$
Q12 Proof Deduction or Consequence from Prior Results View
Let $n \in \mathbb { N } ^ { * }$. Verify that
$$\frac { 2 n \ln ( 2 ) } { \ln ( 2 n ) } - 1 \geqslant \frac { n \ln ( 2 ) } { \ln ( 2 n ) }$$
then deduce that
$$\pi ( 2 n ) \geqslant n \frac { \ln ( 2 ) } { \ln ( 2 n ) }$$
Q13 Proof Bounding or Estimation Proof View
Let $x \geqslant 3$. Show that
$$\pi ( x ) \geqslant \frac { \ln ( 2 ) } { 6 } \frac { x } { \ln ( x ) }$$
One may set $n = \lfloor x / 2 \rfloor$ and use Q12.
Q14 Number Theory GCD, LCM, and Coprimality View
Let $r \in \mathbb { N } ^ { \star }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Justify that there exists a unique natural integer $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ such that
$$a _ { 1 } \mathbb { Z } \cap a _ { 2 } \mathbb { Z } \cap \cdots \cap a _ { r } \mathbb { Z } = d \left( a _ { 1 } , \ldots , a _ { r } \right) \mathbb { Z }$$
Q15 Number Theory GCD, LCM, and Coprimality View
Let $r \in \mathbb { N } ^ { * }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Show that $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ is the smallest non-zero natural integer that is divisible by $a _ { 1 } , \ldots , a _ { r }$.
Q16 Number Theory GCD, LCM, and Coprimality View
Calculate $d _ { 2 } , d _ { 3 }$ and $d _ { 4 }$, then show that $d _ { n } \leqslant n !$ for all natural integer $n \in \mathbb { N } ^ { * }$.
Q17 Number Theory GCD, LCM, and Coprimality View
For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
Show that $d _ { n } = \prod _ { \substack { p \leqslant n \\ p \text { prime } } } p ^ { k _ { p } }$.
Q18 Number Theory GCD, LCM, and Coprimality View
For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
For all prime number $p$, show that $k _ { p } = \left\lfloor \frac { \ln ( n ) } { \ln ( p ) } \right\rfloor$. Deduce that $d _ { n } \leqslant n ^ { \pi ( n ) }$.
Q19 Number Theory Prime Counting and Distribution View
For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$. Deduce that there exists a non-zero natural integer $N$ such that, for all $n \geqslant N , d _ { n } \leqslant 3 ^ { n }$.
One may use the prime number theorem: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.
Q20 Number Theory Irrationality and Transcendence Proofs View
Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$
We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$.
Show that $\alpha$ is an irrational number.
Q21 Number Theory Irrationality and Transcendence Proofs View
Let $\beta = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { 10 ^ { n ! } }$.
Justify that $\beta$ is well-defined, then show that $\beta$ is an irrational number.
Q22 Number Theory Irrationality and Transcendence Proofs View
Let $n \in \mathbb { N } ^ { * }$. Justify that $\zeta ( 2 ) = \sum _ { k = 1 } ^ { + \infty } \frac { 1 } { k ^ { 2 } }$ is well-defined, then show that we can write
$$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { 2 } } = \frac { p _ { n } } { q _ { n } }$$
with $p _ { n } \in \mathbb { N } ^ { * }$ and $q _ { n } = d _ { n } ^ { 2 }$.
Q23 Number Theory Irrationality and Transcendence Proofs View
Can we apply the result of Q20 to these sequences $\left( p _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ and $\left( q _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ to conclude about the irrationality of $\zeta ( 2 )$ ?
Q24 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$.
Let $y \in ] 0,1 [$. Justify that the function
$$x \mapsto \frac { x ^ { r } y ^ { s } } { 1 - x y }$$
is integrable on $[ 0,1 ]$.
Q25 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. For $y \in ] 0,1 [$, we set
$$f _ { r , s } ( y ) = \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x$$
Show that $f _ { r , s }$ is continuous and integrable on $] 0,1 [$.
Q26 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. We set
$$J _ { r , s } = \int _ { 0 } ^ { 1 } f _ { r , s } ( y ) \mathrm { d } y = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
Show that
$$J _ { r , s } = \sum _ { k = 0 } ^ { + \infty } \frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) }$$
Q27 Polynomial Division & Manipulation View
Let $r$ and $s$ be two strictly positive natural integers such that $r > s$. Justify that
$$\frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) } = \frac { 1 } { r - s } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right) .$$
Q28 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \sum _ { k = 0 } ^ { + \infty } \frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) }$$
Deduce that
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = 0 } ^ { + \infty } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right)$$
Q29 Sequences and Series Evaluation of a Finite or Infinite Sum View
Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = 0 } ^ { + \infty } \left( \frac { 1 } { s + k + 1 } - \frac { 1 } { r + k + 1 } \right)$$
Deduce that
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$
Q30 Number Theory GCD, LCM, and Coprimality View
Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$
Deduce that we can write
$$J _ { r , s } = \frac { p _ { r , s } } { q _ { r , s } }$$
with $p _ { r , s }$ and $q _ { r , s }$ natural integers and $q _ { r , s }$ dividing $d _ { r } ^ { 2 }$.
Q31 Factor & Remainder Theorem Polynomial Degree and Structural Properties View
We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
Let $n \in \mathbb { N } ^ { * }$. Justify that $P _ { n }$ is a polynomial function on $[ 0,1 ]$ of degree $n$ with coefficients in $\mathbb { Z }$.
Q32 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We set
$$P _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }, \quad ( 1 - y ) ^ { n } = \sum _ { k = 0 } ^ { n } b _ { k } y ^ { k }$$
with for all $k \in \llbracket 0 , n \rrbracket , a _ { k } \in \mathbb { Z }$ and $b _ { k } \in \mathbb { Z }$.
Let $n \in \mathbb { N } ^ { * }$. Justify the existence of
$$I _ { n } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { ( 1 - y ) ^ { n } P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
and show that
$$I _ { n } = \sum _ { \substack { r , s = 0 \\ r \neq s } } ^ { n } a _ { r } b _ { s } J _ { r , s } + \sum _ { r = 0 } ^ { n } a _ { r } b _ { r } J _ { r , r }$$
Q33 Sequences and Series Functional Equations and Identities via Series View
We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We admit that $J _ { r , r } = \zeta ( 2 ) - \sum _ { k = 1 } ^ { r } \frac { 1 } { k ^ { 2 } }$.
Let $n \in \mathbb { N } ^ { * }$. Deduce that there exist two integers $p _ { n }$ and $q _ { n }$ such that
$$I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$$
Q34 Integration by Parts Prove an Integral Identity or Equality View
We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
Let $n \in \mathbb { N } ^ { * }$. Show that for all $y \in ] 0,1 [$,
$$\int _ { 0 } ^ { 1 } \frac { P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x = ( - y ) ^ { n } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x$$
Q35 Integration by Parts Prove an Integral Identity or Equality View
We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We set $I _ { n } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { ( 1 - y ) ^ { n } P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$.
Deduce that
$$I _ { n } = ( - 1 ) ^ { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } y ^ { n } ( 1 - y ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x \mathrm {~d} y$$
Q36 Stationary points and optimisation Prove an inequality using calculus-based optimisation View
Show that
$$\forall ( x , y ) \in ] 0,1 \left[ ^ { 2 } , \quad \frac { x ( 1 - x ) y ( 1 - y ) } { 1 - x y } \leqslant \frac { 5 \sqrt { 5 } - 11 } { 2 } \right.$$
Q37 Proof Bounding or Estimation Proof View
We have
$$I _ { n } = ( - 1 ) ^ { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } y ^ { n } ( 1 - y ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x \mathrm {~d} y$$
Let $n \in \mathbb { N } ^ { * }$. Deduce that
$$\left| I _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 \sqrt { 5 } - 11 } { 2 } \right) ^ { n }$$
Q38 Proof Existence Proof View
We have $I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$ where $p _ { n }$ and $q _ { n }$ are non-zero integers for all $n \in \mathbb { N } ^ { * }$.
Show that there exists $N \in \mathbb { N } ^ { * }$ such that for all $n \geqslant N$,
$$0 < \left| p _ { n } + \zeta ( 2 ) q _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 } { 6 } \right) ^ { n }$$
One may use, without proving it, the inequality $9 \frac { 5 \sqrt { 5 } - 11 } { 2 } \leqslant \frac { 5 } { 6 }$.
Q39 Proof True/False Justification View
Show that $\zeta ( 2 )$ is an irrational number.
Q40 Proof Deduction or Consequence from Prior Results View
We admit, only in this question, that $\zeta ( 2 ) = \frac { \pi ^ { 2 } } { 6 }$. Show that $\pi$ is an irrational number.