Olympiad problems

1966 IMO Longlists, 26

Prove the inequality [b]a.)[/b] $ \left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left( a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $ where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers. [b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality \[ a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left( a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) }, \] then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.

2011 Armenian Republican Olympiads, Problem 1

Tags: algebra , function

Does there exist a function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any $x>y,$ it satisfies $f(x)-f(y)>\sqrt{x-y}.$

2019 CMI B.Sc. Entrance Exam, 3

Tags: recursion of integrals , definite integral , algebra

Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $

1971 Bulgaria National Olympiad, Problem 2

Tags: equation , algebra

Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

2000 Romania Team Selection Test, 2

Tags: polynomial , algebra

Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$. [i]Marius Cavachi[/i]

2020 Peru Cono Sur TST., P5

Tags: algebra

Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that: $$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$

2022 Assara - South Russian Girl's MO, 6

Tags: algebra , sequence , recurrence relation

There are $2022$ numbers arranged in a circle $a_1, a_2, . . ,a_{2022}$. It turned out that for any three consecutive $a_i$, $a_{i+1}$, $a_{i+2}$ the equality $a_i =\sqrt2 a_{i+2} - \sqrt3 a_{i+1}$. Prove that $\sum^{2022}_{i=1} a_ia_{i+2} = 0$, if we know that $a_{2023} = a_1$, $a_{2024} = a_2$.

1969 IMO Longlists, 66

Tags: algebra , inequality , taylor series , polynomial

$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$ $(b)$ Formulate and prove the analogous result for polynomials of third degree.

2007 Hungary-Israel Binational, 3

Tags: polynomial , algebra

Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.

2017 IFYM, Sozopol, 5

Tags: algebra , functional equation

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation $f(xy+x+y)=f(xy)+f(x)+f(y)$ is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.

2010 Contests, 3

Tags: algebra , perfect square , function , number theory

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2005 Czech And Slovak Olympiad III A, 1

Tags: arithmetic sequence , algebra , sequence , maximum , recurrence relation

Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.

2011 Baltic Way, 2

Tags: functional equation in n , algebra , functional equation

Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds: \[f(f(x)-y)=f(y)-f(f(x)).\] Show that $f$ is bounded.

1969 All Soviet Union Mathematical Olympiad, 117

Tags: algebra , digit , sequence

Given a finite sequence of zeros and ones, which has two properties: a) if in some arbitrary place in the sequence we select five digits in a row and also select five digits in any other place in a row, then these fives will be different (they may overlap); b) if you add any digit to the right of the sequence, then property (a) will no longer hold true. Prove that the first four digits of our sequence coincide with the last four

2018 Mathematical Talent Reward Programme, MCQ: P 5

Tags: algebra , trigonometry

Let the maximum and minimum value of $f(x)=\cos \left(x^{2018}\right) \sin x$ are $M$ and $m$ respectively where $x \in[-2 \pi, 2 \pi] .$ Then $$ M+m= $$ [list=1] [*] $\frac{1}{2}$ [*] $-\frac{1}{\sqrt{2}}$ [*] $\frac{1}{2018}$ [*] Does not exists [/list]

2024 Middle European Mathematical Olympiad, 2

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\] for all $x,y \in \mathbb{R}$.

2005 Postal Coaching, 19

Tags: function , polynomial , algebra

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

2024 Switzerland Team Selection Test, 3

Tags: algebra , polynomial

Determine all monic polynomial with integer coefficient $P$ such that for every integer $a,b$ there exists integer $c$ so that \[P(a)P(b)=P(c)\]

2014 Bulgaria JBMO TST, 6

Tags: algebra

If $a,b$ are real numbers such that $a^3 +12a^2 + 49a + 69 = 0$ and $ b^3 - 9b^2 + 28b - 31 = 0,$ find $a + b .$

2007 Romania National Olympiad, 2

Tags: logarithm , algebra

Solve the equation \[2^{x^{2}+x}+\log_{2}x = 2^{x+1}\]

1998 Yugoslav Team Selection Test, Problem 3

Tags: algebra , sequence

Prove that there are no positive integers $n$ and $k\le n$ such that the numbers $$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.

2007 Indonesia TST, 3

Tags: number theory , induction , function , algebra , polynomial

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

1991 Arnold's Trivium, 45

Tags: algebra , polynomial , projective geometry

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.

2015 All-Russian Olympiad, 3

Tags: algebra , inequalities , number theory

Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.

1992 IMO Shortlist, 19

Tags: coefficient , divisibility , polynomial , algebra , number theory

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$