Olympiad problems

2025 CMIMC Algebra/NT, 6

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1].$ Find the probability that $$|x|+|y|+1 \le 3\min\{|x+y+1|, |x+y-1|\}.$$

2017 Thailand Mathematical Olympiad, 3

Tags: functional equation , functional , algebra

Determine all functions $f : R \to R$ satisfying $f(f(x) - y) \le xf(x) + f(y)$ for all real numbers $x, y$.

III Soros Olympiad 1996 - 97 (Russia), 11.2

Tags: inequalities , trigonometry , algebra

Find the smallest value of the expression: $$y=\frac{x^2}{8}+x \cos x +\cos 2x$$

2014 Contests, 2

Tags: algebra , function , functional equation

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

1993 Tournament Of Towns, (371) 3

Tags: pascal triangle , algebra

Each number in the second, third, and further rows of the following triangle: [img]https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png[/img] is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle). (GW Leibnitz,)

II Soros Olympiad 1995 - 96 (Russia), 11.4

Tags: polynomial , algebra

Prove that the equation $x^6 - 100x+1 = 0$ has two roots, and both of these roots are positive. a) Find the first non-zero digit in the decimal notation of the lesser root of this equation. b) Find the first two non-zero digits in the decimal notation of the lesser root of this equation.

2000 Romania National Olympiad, 3

Tags: algebra

Let be a natural number $ n\ge 2 $ and an expression of $ n $ variables $$ E\left( x_1,x_2,...,x_n\right) =x_1^2+x_2^2+\cdots +x_n^2-x_1x_2-x_2x_3-\cdots -x_{n-1}x_n -x_nx_1. $$ Determine $ \sup_{x_1,...,x_n\in [0,1]} E\left( x_1,x_2,...,x_n\right) $ and the specific values at which this supremum is attained.

2010 Kosovo National Mathematical Olympiad, 5

Tags: inequalities , inequality , algebra

Let $x,y$ be positive real numbers such that $x+y=1$. Prove that $\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$.

Kvant 2019, M2562

Tags: functional equation , algebra

Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

1998 VJIMC, Problem 2

Tags: arithmetic sequence , palindrome , number theory , algebra

Decide whether there is a member in the arithmetic sequence $\{a_n\}_{n=1}^\infty$ whose first member is $a_1=1998$ and the common difference $d=131$ which is a palindrome (palindrome is a number such that its decimal expansion is symmetric, e.g., $7$, $33$, $433334$, $2135312$ and so on).

2003 Czech And Slovak Olympiad III A, 3

Tags: recurrence relation , recurrence , algebra , combinatorics

A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.

2021 CMIMC, 2.8 1.4

Tags: algebra

Let $f(x) = \frac{x^2}8$. Starting at the point $(7,3)$, what is the length of the shortest path that touches the graph of $f$, and then the $x$-axis? [i]Proposed by Sam Delatore[/i]

2008 Moldova National Olympiad, 12.5

Tags: polynomial , algebra

Find the least positive integer $ n$ so that the polynomial $ P(X)\equal{}\sqrt3\cdot X^{n\plus{}1}\minus{}X^n\minus{}1$ has at least one root of modulus $ 1$.

2008 ISI B.Math Entrance Exam, 8

Tags: quadratic , trigonometry , algebra

Let $a^2+b^2=1$ , $c^2+d^2=1$ , $ac+bd=0$ Prove that $a^2+c^2=1$ , $b^2+d^2=1$ , $ab+cd=0$ .

2019 Saudi Arabia BMO TST, 2

Tags: sequence , recurrence relation , algebra

Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$ and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $

2003 IMO Shortlist, 2

Tags: functional equation , function , algebra

Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$. [i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]

2023 JBMO TST - Turkey, 3

Tags: function , algebra

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$

2015 Harvard-MIT Mathematics Tournament, 9

Tags: divisibility , algebra , number theory

Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

2015 Iran MO (3rd round), 3

Tags: function , algebra , polynomial

Does there exist an irreducible two variable polynomial $f(x,y)\in \mathbb{Q}[x,y]$ such that it has only four roots $(0,1),(1,0),(0,-1),(-1,0)$ on the unit circle.

2001 Spain Mathematical Olympiad, Problem 1

Tags: geometry , algebra , polynomial

Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that: $P(x) = b + (x-a)Q((x-a)^2)).$

1991 Tournament Of Towns, (306) 3

Tags: algebra , combinatorics

Is it possible to put pairwise distinct positive integers less than $100$ in the cells of a $4 \times 4$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow))

1984 Putnam, A3

Tags: linear algebra , limit , matrix , algebra , polynomial

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by $m_{ij}=x$ if $i=j$, $m_{ij}=a$ if $i \not= j$ and $i+j$ is even, $m_{ij}=b$ if $i \not= j$ and $i+j$ is odd. For example $ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}$. Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ . P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)

1999 All-Russian Olympiad Regional Round, 9.3

Tags: inequalities , algebra

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$ then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

1953 AMC 12/AHSME, 44

Tags: quadratic , algebra , polynomial

In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $ 8$ and $ 2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $ \minus{}9$ and $ \minus{}1$ for the roots. The correct equation was: $ \textbf{(A)}\ x^2\minus{}10x\plus{}9\equal{}0 \qquad\textbf{(B)}\ x^2\plus{}10x\plus{}9\equal{}0 \qquad\textbf{(C)}\ x^2\minus{}10x\plus{}16\equal{}0\\ \textbf{(D)}\ x^2\minus{}8x\minus{}9\equal{}0 \qquad\textbf{(E)}\ \text{none of these}$

2006 Germany Team Selection Test, 2

Tags: inequalities , algebra , calculus

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.