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

1979 IMO, 2
Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

1986 IMO Shortlist, 5
Tags: quadratic , modular arithmetic , number theory , system of equations , perfect square
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

2023 Israel National Olympiad, P4
Tags: algebra , system of equations , parameterization
For each positive integer $n$, find all triples $a,b,c$ of real numbers for which \[\begin{cases}a=b^n+c^n\\ b=c^n+a^n\\ c=a^n+b^n\end{cases}\]

1997 Tuymaada Olympiad, 2
Tags: diophantine equation , algebra , system of equations , natural number
Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

1995 IMO Shortlist, 4
Tags: function , inequalities , optimization , system of equations , 133
Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

III Soros Olympiad 1996 - 97 (Russia), 11.3
Tags: algebra , system of equations
Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.

1988 All Soviet Union Mathematical Olympiad, 484
Tags: trigonometry , system of equations , minimum , algebra , sum
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?

2015 German National Olympiad, 1
Tags: equation , system of equations , algebra
Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10,\\ y^3+xy^2 &=2. \end{align*}

III Soros Olympiad 1996 - 97 (Russia), 9.4
Tags: algebra , system of equations
Solve the system of equations $$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$

1994 Tournament Of Towns, (433) 3
Tags: algebra , system of equations
Let $a, b, c$ and $d$ be real numbers such that $$a^3+b^3+c^3+d^3=a+b+c+d=0$$ Prove that the sum of a pair of these numbers is equal to $0$. (LD Kurliandchik)

2004 Denmark MO - Mohr Contest, 4
Tags: algebra , system of equations , system
Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$

2008 Mathcenter Contest, 1
Tags: system of equations , system , algebra
Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations : $$x^2+xy+y^2=57$$ $$y^2+yz+z^2=84$$ $$z^2+zx+x^2=111$$ What is the value of $xy+3yz+5zx$? [i](maphybich)[/i]

1994 AIME Problems, 7
Tags: geometry , quadratic , analytic geometry , algebra , system of equations , linear equation
For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

2023 HMNT, 7
Tags: system of equations , algebra
Compute all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations: $$xy + z = 40$$ $$xz + y = 51$$ $$x + y + z = 19.$$

2017 QEDMO 15th, 7
Tags: algebra , system of equations , system
Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$

2004 Cuba MO, 1
Tags: algebra , system of equations , system
Determine all real solutions to the system of equations: $$x_1 + x_2 +...+ x_{2004 }= 2004$$ $$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$

2016 Dutch BxMO TST, 2
Tags: algebra , system of equations
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

2016 Irish Math Olympiad, 8
Tags: algebra , system of equations , algebraic identities
Suppose $a, b, c$ are real numbers such that $abc \ne 0$. Determine $x, y, z$ in terms of $a, b, c$ such that $bz + cy = a, cx + az = b, ay + bx = c$. Prove also that $\frac{1 - x^2}{a^2} = \frac{1 - y^2}{b^2} = \frac{1 - z^2}{c^2}$.

2013 F = Ma, 14
Tags: algebra , system of equations
A cart of mass $m$ moving at $12 \text{ m/s}$ to the right collides elastically with a cart of mass $4.0 \text{ kg}$ that is originally at rest. After the collision, the cart of mass $m$ moves to the left with a velocity of $6.0 \text{ m/s}$. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass ($v_{\text{cm}}$) of the two carts before the collision? $\textbf{(A) } v_{\text{cm}} = 2.0 \text{ m/s}\\ \textbf{(B) } v_{\text{cm}}=3.0 \text{ m/s}\\ \textbf{(C) } v_{\text{cm}}=6.0 \text{ m/s}\\ \textbf{(D) } v_{\text{cm}}=9.0 \text{ m/s}\\ \textbf{(E) } v_{\text{cm}}=18.0 \text{ m/s}$

II Soros Olympiad 1995 - 96 (Russia), 9.6
Tags: system of equations , inequalities , algebra , analytic geometry
Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

2000 German National Olympiad, 1
Tags: system of equations , parameter , algebra
For each real parameter $a$, find the number of real solutions to the system $$\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}$$

1979 Romania Team Selection Tests, 3.
Tags: inequalities , simultaneous equation , algebra , system of equations
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \] we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. [i]Radu Gologan[/i]

2018 CMIMC Algebra, 5
Tags: algebra , system of equations
Suppose $a$, $b$, and $c$ are nonzero real numbers such that \[bc+\frac1a = ca+\frac2b = ab+\frac7c = \frac1{a+b+c}.\] Find $a+b+c$.

2024 Nigerian MO Round 2, Problem 2
Tags: algebra , system of equations
Solve the system of equations: \[x>y>z\] \[x+y+z=1\] \[x^2+y^2+z^2=69\] \[x^3+y^3+z^3=271\] [hide=Answer]x=7, y=-2, z=-4[/hide]

2011 Hanoi Open Mathematics Competitions, 7
Tags: algebra , system of equations
Find all pairs $(x, y)$ of real numbers satisfying the system : $\begin{cases} x + y = 3 \\ x^4 - y^4 = 8x - y \end{cases}$