Olympiad problems

1964 Kurschak Competition, 3

Tags: inequalities , algebra

Show that for any positive reals $w, x, y, z$ we have $$\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}$$

2003 IMO Shortlist, 1

Tags: linear algebra , tetrahedron , geometry , algebra , 3d geometry , vector

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

2015 Finnish National High School Mathematics Comp, 1

Tags: equation , algebra , radical

Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.

2025 Romania National Olympiad, 4

Tags: algebra , number theory , prime number , finite field , polynomial

Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

2010 HMNT, 5

Tags: algebra

A polynomial $P$ is of the form $\pm x^6 \pm x^5 \pm x^4 \pm x^3 \pm x^2 \pm x \pm 1$. Given that $P(2) = 27$, what is $P(3)$?

2020 Brazil Cono Sur TST, 3

Tags: number theory , algebra , sequence

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

1993 China Team Selection Test, 2

Tags: inequalities , algebra , function , quadratic

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

1992 India Regional Mathematical Olympiad, 1

Tags: polynomial , algebra , inequalities

Determine the set of integers $n$ for which $n^2+19n+92$ is a square.

2017 Taiwan TST Round 2, 1

Tags: functional equation , algebra

Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $

2017 Greece Junior Math Olympiad, 2

Tags: function , algebra , inequalities

Let $x,y,z$ is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}$

2021 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Find all real numbers $a,b,c$ and $d$ such that: $a^2+b^2+c^2+d^2=a+b+c+d-ab=3.$

2020 Regional Olympiad of Mexico Northeast, 1

Tags: recurrence relation , algebra , floor function , sequence

Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$? Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.

1998 Polish MO Finals, 1

Tags: algebra

Find all solutions in positive integers to: \begin{eqnarray*} a + b + c = xyz \\ x + y + z = abc \end{eqnarray*}

2017 NMTC Junior, 1

Tags: number theory , algebra

(a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes. (b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\]

2003 Mediterranean Mathematics Olympiad, 3

Tags: algebra , inequalities

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

2014 Online Math Open Problems, 22

Tags: polynomial , function , algebra , modular arithmetic

Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$. [i]Proposed by Michael Kural and Evan Chen[/i]

2008 German National Olympiad, 3

Tags: function , algebra

Find all functions $ f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $ x$ it is $ f(x) \geq 0$. (ii) It is $ f\left(1\right)\equal{}\frac 12$. (iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.

1997 Tuymaada Olympiad, 5

Tags: algebra , inequalities

Prove the inequality $\left(1+\frac{1}{q}\right)\left(1+\frac{1}{q^2}\right)...\left(1+\frac{1}{q^n}\right)<\frac{q-1}{q-2}$ for $n\in N, q>2$

2021 BMT, 4

Tags: trigonometry , algebra

Let $\theta$ be a real number such that $1 + \sin 2\theta -\left(\frac12 \sin 2\theta\right)^2= 0$. Compute the maximum value of $(1 + \sin \theta )(1 + \cos \theta)$.

2004 Bulgaria Team Selection Test, 3

Tags: number theory , algebra

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

2019 Paraguay Mathematical Olympiad, 1

Tags: algebra , trinomial , quadratic

Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

1990 Baltic Way, 11

Tags: polynomial , algebra

Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.

2009 Poland - Second Round, 3

Tags: algebra , vector , linear algebra

For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.

2021 Belarusian National Olympiad, 10.1

Tags: algebra , limit , sequence

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

1991 Turkey Team Selection Test, 3

Tags: algebra , polynomial , function

Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.