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

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 $

1967 IMO Shortlist, 5
Tags: trigonometry , algebra , system of equations , trigonometric equation
If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

V Soros Olympiad 1998 - 99 (Russia), 9.3
Tags: system of equations , fractional part , algebra
Solve the system of equations: $$\begin{cases} x + [y] + \{z\}=3.9 \\ y + [z] + \{x\}= 3.5 \\ z + [x] + \{y\}= 2. \end{cases}$$

2022 Polish MO Finals, 4
Tags: algebra , system of equations
Find all triples $(a,b,c)$ of real numbers satisfying the system $\begin{cases} a^3+b^2c=ac \\ b^3+c^2a=ba \\ c^3+a^2b=cb \end{cases}$

2022 Chile Junior Math Olympiad, 1
Tags: system of equations , algebra
Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

2022 Cyprus TST, 1
Tags: system of equations , algebra
Find all pairs of real numbers $(x,y)$ for which \[ \begin{aligned} x^2+y^2+xy&=133 \\ x+y+\sqrt{xy}&=19 \end{aligned} \]

1966 IMO, 5
Tags: algebra , system of equations
Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

1976 Swedish Mathematical Competition, 2
Tags: algebra , system of equations , system
For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$

2004 Nicolae Coculescu, 1
Tags: equation , system of equations , algebra , monotone
Solve in the real numbers the system: $$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$ [i]Eduard Buzdugan[/i]

1965 IMO Shortlist, 2
Tags: linear algebra , matrix , function , algebra , system of equations
Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

1952 Moscow Mathematical Olympiad, 222
Tags: algebra , system of equations
a) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{14}x_{15} = 0 \\ 1 - x_{15}x_1 = 0 \end{cases}$ b) Solve the system of equations $\begin{cases} 1 - x_1x_2 = 0 \\ 1 - x_2x_3 = 0 \\ ...\\ 1 - x_{n-1}x_{n} = 0 \\ 1 - x_{n}x_1 = 0 \end{cases}$ How does the solution vary for distinct values of $n$?

2010 Saudi Arabia Pre-TST, 2.1
Tags: diophantine , system of equations , system , number theory
Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$

2006 China Second Round Olympiad, 3
Tags: system of equations , algebra
Solve the system of equations in real numbers: \[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]

2017 Finnish National High School Mathematics Comp, 2
Tags: algebra , system of equations
Determine $x^2+y^2$ and $x^4+y^4$, when $x^3+y^3=2$ and $x+y=1$

2005 Denmark MO - Mohr Contest, 5
Tags: algebra , system of equations , system
For what real numbers $p$ has the system of equations $$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$ just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?

2022 Bulgarian Spring Math Competition, Problem 10.4
Tags: algebra , system of equations , prime
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

2011 Indonesia TST, 1
Tags: algebra , system of equations
Find all $4$-tuple of real numbers $(x, y, z, w)$ that satisfy the following system of equations: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

2019-IMOC, A3
Tags: system of equations , algebra
Find all $3$-tuples of positive reals $(a,b,c)$ such that $$\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}$$

2000 Austrian-Polish Competition, 3
Tags: algebra , system of equations
For each integer $n \ge 3$ solve in real numbers the system of equations: $$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$

2023-IMOC, N6
Tags: algebra , system of equations
Let $S(b)$ be the number of nonuples of positive integers $(a_1, a_2, \ldots , a_9)$ satisfying $3b-1=a_1+a_2+\ldots+a_9$ and $b^2+1=a_1^2+\ldots+a_9^2$. Prove that for all $\epsilon>0$, there exists $C_{\epsilon}>0$ such that $S(b)\leq C_{\epsilon}b^{3+\epsilon}$.

2023 Moldova EGMO TST, 5
Tags: equation , system of equations
Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$

2015 Lusophon Mathematical Olympiad, 4
Tags: algebra , system of equations
Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers . Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.

2004 Vietnam National Olympiad, 1
Tags: system of equations , algebra
Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.

2012 Purple Comet Problems, 7
Tags: algebra , system of equations
Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.

2024 Argentina National Math Olympiad Level 3, 1
Tags: system of equations , algebra
Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations: $$\left \{\begin{matrix} a\cdot b+c+d & = & 6, \\ b\cdot c+d+a & = & 2, \\ c\cdot d+a+b & = & 5, \\ d\cdot a+b+c & = & 3. \end{matrix}\right .$$