This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 744

2018 Dutch Mathematical Olympiad, 3
Tags: algebra , system of equations
Determine all triples $(x, y,z)$ consisting of three distinct real numbers, that satisfy the following system of equations: $\begin {cases}x^2 + y^2 = -x + 3y + z \\ y^2 + z^2 = x + 3y - z \\ x^2 + z^2 = 2x + 2y - z \end {cases}$

2014 Hanoi Open Mathematics Competitions, 9
Tags: algebra , system of equations
Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\ 16y^3 + 4y = 16x + 5 \end{cases}$

2007 Puerto Rico Team Selection Test, 2
Tags: number theory , diophantine equation , diophantine , algebra , system of equations
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

2021 Dutch IMO TST, 2
Tags: system of equations , system , algebra
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2015 BMT Spring, 3
Tags: algebra , system of equations , number theory
Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

2014 AMC 12/AHSME, 16
Tags: algebra , polynomial , system of equations
Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$? $ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $

2017 Germany, Landesrunde - Grade 11/12, 6
Tags: system of equations , algebra
Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right), \\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right). \end{align*}

2020 Junior Balkаn MO, 1
Tags: algebra , system of equations , polynomial
Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2010 Mediterranean Mathematics Olympiad, 1
Tags: system of equations , algebra
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

2023 German National Olympiad, 4
Tags: system of equations , cyclic system , algebra
Determine all triples $(a,b,c)$ of real numbers with \[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]

1979 All Soviet Union Mathematical Olympiad, 276
Tags: algebra , system of equations
Find $x$ and $y$ ($a$ and $b$ parameters): $$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$

1999 Czech And Slovak Olympiad IIIA, 6
Tags: system of equations , algebra
Find all pairs of real numbers $a,b$ for which the system of equations $$ \begin{cases} \dfrac{x+y}{x^2 +y^2} = a \\ \\ \dfrac{x^3 +y^3}{x^2 +y^2} = b \end{cases}$$ has a real solution.

1980 All Soviet Union Mathematical Olympiad, 292
Tags: trigonometry , system of equations , algebra
Find real solutions of the system : $$\begin{cases} \sin x + 2 \sin (x+y+z) = 0 \\ \sin y + 3 \sin (x+y+z) = 0\\ \sin z + 4 \sin (x+y+z) = 0\end{cases}$$

1967 Putnam, A6
Tags: system of equations , linear equation
Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations $$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$ for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$). [list=a] [*] Determine, with proof, the maximum number of distinct $4$-tuples possible. [*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.

2002 AIME Problems, 6
Tags: logarithm , quadratic , algebra , system of equations
The solutions to the system of equations \begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*} are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$

2010 Junior Balkan Team Selection Tests - Romania, 3
Tags: system of equations , algebra
Let $n \ne 0$ be a natural number and integers $x_1, x_2, ...., x_n, y_1, y_2, ...., y_n$ with the properties: a) $x_1 + x_2 + .... + x_n = y_1 + y_2 + .... + y_n = 0,$ b) $x_1 ^ 2 + y_1 ^ 2 = x_2 ^ 2 + y_2 ^ 2 = .... = x_n ^ 2 + y_n ^ 2$. Show that $n$ is even.

2024 Kyiv City MO Round 2, Problem 1
Tags: algebra , system of equations
Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$ [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

1962 Poland - Second Round, 1
Tags: algebra , system of equations
Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw $$x + y + z = a,$$ $$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$ then at least one of them is equal to $ a $.

1991 IMO, 2
Tags: number theory , arithmetic sequence , system of equations , eulers function , laurentiu panaitopol
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

2023 Greece National Olympiad, 1
Tags: algebra , system of equations
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system: $\begin{cases} \frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\ x+y+z+w= 48 \end{cases}$

1989 Romania Team Selection Test, 2
Tags: system of equations , algebra , number theory , coprime
Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

2016 Regional Olympiad of Mexico West, 5
Tags: algebra , system of equations
Determine all real solutions of the following system of equations: $$x+y^2=y^3$$ $$y+x^2=x^3$$

2010 Contests, 1
Tags: pigeonhole principle , system of equations , algebra
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

III Soros Olympiad 1996 - 97 (Russia), 10.9
Tags: system of equations , algebra
For any positive $a$ and $b$, find positive solutions of the system $$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4) \\ ax-by=x^4-y^4 \end{cases}$$

2025 Poland - First Round, 8
Tags: algebra , system of equations
Real numbers $a, b, c, x, y, z$ satisfy $$\begin{aligned} \begin{cases} a^2+2bc=x^2+2yz,\\ b^2+2ca=y^2+2zx,\\ c^2+2ab=z^2+2xy.\\ \end{cases} \end{aligned}$$ Prove that $a^2+b^2+c^2=x^2+y^2+z^2$.