Olympiad problems

2009 IMAR Test, 1

Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

2019 Czech and Slovak Olympiad III A, 1

Tags: algebra , system of equations

Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}

2008 Czech-Polish-Slovak Match, 1

Tags: system of equations , algebra

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations \[2x^3=2y(x^2+1)-(z^2+1), \] \[ 2y^4=3z(y^2+1)-2(x^2+1), \] \[ 2z^5=4x(z^2+1)-3(y^2+1).\]

1990 China National Olympiad, 4

Tags: algebra , system of equations , number theory

Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution: \[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]

2014 India Regional Mathematical Olympiad, 4

Tags: number theory , least common multiple , inequalities , symmetry , vieta , algebra , system of equations

Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]

2004 Germany Team Selection Test, 1

Tags: polynomial , linear algebra , matrix , complex number , system of equations , algebra

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

2020 Bundeswettbewerb Mathematik, 2

Tags: rational number , algebra , system of equations , number theory

Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.

2018 Dutch IMO TST, 1

Tags: system of equations , algebra

(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold? (b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers, does $a = b = c$ necessarily hold?

1926 Eotvos Mathematical Competition, 1

Tags: diophantine , diophantine equation , system of equations , system , number theory

Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

1967 IMO Shortlist, 6

Tags: linear algebra , matrix , algebra , system of equations

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

2017 India PRMO, 18

Tags: algebra , system of equations

If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?

2018-IMOC, A7

Tags: algebra , system of equations

If the reals $a,b,c,d,e,f,g,h,i$ satisfy $$\begin{cases}ab+bc+ca=3\\de+ef+fd=3\\gh+hi+ig=3\\ad+dg+ga=3\\be+eh+hb=3\end{cases}$$show that $cf+fi+ic=3$ holds as well.

1993 Austrian-Polish Competition, 5

Tags: system of equations , algebra , austrian polish

Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}$$

1972 Poland - Second Round, 1

Tags: algebra , system of equations , system

Prove that there are no real numbers $ a, b, c $, $ x_1, x_2, x_3 $ such that for every real number $ x $ $$ ax^2 + bx + c = a(x - x_2)(x - x_3) $$ $$bx^2 + cx + a = b(x - x_3) (x - x_1)$$ $$cx^2 + ax + b = c(x - x_1) (x - x_2)$$ and $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.

1990 Austrian-Polish Competition, 3

Tags: algebra , system of equations

Show that there are two real solutions to: $$\begin{cases} x + y^2 + z^4 = 0 \\ y + z^2 + x^4 = 0 \\ z + x^2 + y^5 = 0\end {cases}$$

2024 AIME, 4

Tags: algebra , system of equations , number theory , relatively prime

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

1968 German National Olympiad, 1

Tags: algebra , system of equations , system

Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1 + ax_2 + x_3 = b $$ $$x_2 + ax_3 + x_4 = b $$ $$x_3 + ax_4 + x_1 = b $$ $$x_4 + ax_1 + x_2 = b$$ Here $a$ and $b$ are real numbers (case distinction!).

2022 IFYM, Sozopol, 1

Tags: algebra , system of equations , system

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

1991 Austrian-Polish Competition, 8

Tags: number theory , system of equations

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

2012 Finnish National High School Mathematics Competition, 2

Tags: system of equations , algebra

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

IV Soros Olympiad 1997 - 98 (Russia), 10.4

Tags: system of equations , algebra

Solve the system of equations $$\begin{cases} x+y+z+t=6 \\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8 \end{cases}$$

III Soros Olympiad 1996 - 97 (Russia), 10.9

Tags: algebra , system of equations

For any positive $a$ and $b$, find positive solutions of the system $$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4) \\ ax-by=x^4-y^4 \end{cases}$$

2018 Latvia Baltic Way TST, P2

Tags: system of equations , algebra

Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations: $$\begin{cases} y(x+y)^2=2\\ 8y(x^3-y^3) = 13. \end{cases}$$

2022 Poland - Second Round, 1

Tags: algebra , system of equations

Find all real quadruples $(a,b,c,d)$ satisfying the system of equations $$ \left\{ \begin{array}{ll} ab+cd = 6 \\ ac + bd = 3 \\ ad + bc = 2 \\ a + b + c + d = 6. \end{array} \right. $$

1966 IMO Shortlist, 58

Tags: combinatorics , system of equations , algebra

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?