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

Tags were heavily modified to better represent problems.

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Found problems: 744

1997 Poland - Second Round, 1
Tags: system of equations , algebra
For the real number $a$ find the number of solutions $(x, y, z)$ of a system of the equations: $\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.$

2001 Estonia National Olympiad, 1
Tags: algebra , system of equations , trigonometry
Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$

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*}

2018 OMMock - Mexico National Olympiad Mock Exam, 3
Tags: algebra , system of equations , algebraic manipulations
Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds: \[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \] [i]Proposed by Oriol Solé[/i]

2004 Switzerland Team Selection Test, 7
Tags: system of equations , product , algebra
The real numbers $a,b,c,d$ satisfy the equations: $$\begin{cases} a =\sqrt{45-\sqrt{21-a}} \\ b =\sqrt{45+\sqrt{21-b}}\\ c =\sqrt{45-\sqrt{21+c}}\ \\ d=\sqrt{45+\sqrt{21+d}} \end {cases}$$ Prove that $abcd = 2004$.

1984 USAMO, 4
Tags: algebra , system of equations , combinatorics
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

2021 Latvia Baltic Way TST, P3
Tags: algebra , system of equations
Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

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

2014 Saudi Arabia IMO TST, 3
Tags: induction , algebra , system of equations , number theory
Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

2019 Greece Junior Math Olympiad, 1
Tags: system of equations , algebra
Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations $x^2+y^2+25z^2=6xz+8yz$ $ 3x^2+2y^2+z^2=240$

VI Soros Olympiad 1999 - 2000 (Russia), 8.5
Tags: number theory , system of equations
Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

2017 Istmo Centroamericano MO, 4
Tags: diophantine equation , number theory , diophantine , system of equations , inequalities
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

2012 India Regional Mathematical Olympiad, 1
Tags: algebra , polynomial , system of equations
Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.

1999 Vietnam National Olympiad, 1
Tags: system of equations , algebra
Solve the system of equations: $ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$ $ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$

1993 Nordic, 3
Tags: system of equations , digit , positive integer , number theory
Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

2019 India Regional Mathematical Olympiad, 3
Tags: algebra , inequality , system of equations
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

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

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

2004 German National Olympiad, 1
Tags: real , algebra , system of equations , equation
Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}

2003 AMC 8, 4
Tags: algebra , system of equations , binomial theorem
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

2006 Denmark MO - Mohr Contest, 2
Tags: algebra , system of equations , system
Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

2023 New Zealand MO, 5
Tags: algebra , system of equations
Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$. Prove that $x + y + z$ is an integer.

2003 German National Olympiad, 1
Tags: system of equations , algebra
Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$

2004 USAMTS Problems, 2
Tags: algebra , system of equations
Find positive integers $a$, $b$, and $c$ such that \[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\] Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)

2012 Greece JBMO TST, 1
Tags: algebra , equation , system of equations
Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$