Olympiad problems

2023 JBMO Shortlist, A5

Tags: algebra , inequality

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2010 Putnam, B6

Tags: polynomial , algebra , matrix , linear algebra , induction , vector

Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$

2017 Taiwan TST Round 1, 1

Tags: algebra , polynomial , functional equation

Find all polynomials $P$ with real coefficients which satisfy \[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]

2020 MMATHS, 5

Tags: algebra , minimum

Let $x, y$ be positive reals such that $x \ne y$. Find the minimum possible value of $(x + y)^2 + \frac{54}{xy(x-y)^2}$ .

2003 Canada National Olympiad, 3

Tags: algebra

Find all real positive solutions (if any) to \begin{align*} x^3+y^3+z^3 &= x+y+z, \mbox{ and} \\ x^2+y^2+z^2 &= xyz. \end{align*}

2024 European Mathematical Cup, 4

Tags: algebra , function , functional equation

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$ for all x, y positive reals.

2025 Kyiv City MO Round 1, Problem 4

Tags: algebra

Oleksii wrote some $ 2n $ ($ n > 1 $) consecutive positive integers on the board. After that, he grouped these numbers into pairs in some way, and within each pair, he multiplied the two numbers together. He then wrote the resulting $ n $ products on the board instead of the original numbers. Afterward, Anton wrote down the difference between the largest and the smallest of the numbers Oleksii wrote. Oleksii wants Anton to write the smallest possible number. What is the smallest number that can be written? [i]Proposed by Oleksii Masalitin, Anton Trygub[/i]

1994 Korea National Olympiad, Problem 1

Tags: algebra , function

Let $ S$ be the set of nonnegative integers. Find all functions $ f,g,h: S\rightarrow S$ such that $ f(m\plus{}n)\equal{}g(m)\plus{}h(n),$ for all $ m,n\in S$, and $ g(1)\equal{}h(1)\equal{}1$.

2021 MOAA, 4

Tags: speed , inequalities , algebra

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

2019 Brazil Undergrad MO, 3

Tags: calculus , algebra , system of equations

Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

1993 AIME Problems, 5

Tags: algebra , polynomial

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

2012 Cuba MO, 1

Tags: algebra , system of equations

If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

1981 Romania Team Selection Tests, 2.

Tags: algebra

Show that a set $A$ consisting of $16$ consecutive non-negative integers can be partitioned in two disjoint sets $X$ and $Y$ each containing $8$ elements so that $\sum\limits_{x\in X}x^k=\sum\limits_{y\in Y} y^k,$ for $k=1,2,3.$

2022 Benelux, 1

Tags: polynomial , algebra

Let $n\geqslant 0$ be an integer, and let $a_0,a_1,\dots,a_n$ be real numbers. Show that there exists $k\in\{0,1,\dots,n\}$ such that $$a_0+a_1x+a_2x^2+\cdots+a_nx^n\leqslant a_0+a_1+\cdots+a_k$$ for all real numbers $x\in[0,1]$.

2008 Mathcenter Contest, 4

Tags: number theory , algebra , integer , rational

Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer. [i](Anonymous314)[/i]

1947 Moscow Mathematical Olympiad, 131

Tags: product , algebra

Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product $\left(1-\frac{1}{10}\right)\left(1-\frac{1}{10^2}\right)...\left(1-\frac{1}{10^{99}}\right)$

1991 India Regional Mathematical Olympiad, 6

Tags: quadratic , quadratic formula , algebra

Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers.

2023 Kazakhstan National Olympiad, 4

Tags: algebra , inequalities

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

2020 Dutch IMO TST, 3

Tags: algebra , functional equation in z , functional , functional equation

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

2016 Iran MO (3rd Round), 1

Tags: polynomial , algebra , number theory

Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold [list] [*] $a+b \in F$, and [*] $\gcd(P(a),P(b))=1$. [/list] Prove that $P(x)$ is a constant polynomial.

2016 Dutch BxMO TST, 2

Tags: system of equations , algebra

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

2008 Romania National Olympiad, 4

Tags: matrix , linear algebra , algebra , polynomial , inequalities

Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]

2009 Romanian Masters In Mathematics, 1

Tags: least common multiple , number theory , triangle inequality , inequalities , greatest common divisor , floor function , algebra

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

2012 Abels Math Contest (Norwegian MO) Final, 1a

Tags: algebra , number theory

Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?

2017 QEDMO 15th, 7

Tags: system , algebra , system of equations

Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$