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

2022 South Africa National Olympiad, 2
Tags: quadratic , algebra , system of equations
Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

1959 Czech and Slovak Olympiad III A, 4
Tags: system of equations , trigonometry
Find all pair $(x, y)$ in degrees such that \begin{align*} &\sin (x + 150^\circ) = \cos (y - 75^\circ), \\ &\cos x + \sin (y - 225^\circ) + \frac{\sqrt3}{2} = 0. \end{align*}

2013 Purple Comet Problems, 30
Tags: algebra , system of equations
Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$

2019 AIME Problems, 3
Tags: algebra , system of equations
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

1982 Austrian-Polish Competition, 7
Tags: diophantine , diophantine equation , system of equations , number theory
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$.

2003 Switzerland Team Selection Test, 1
Tags: system of equations , parameter , algebra
Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$ Find all possible values of $a$.

2009 USA Team Selection Test, 7
Tags: polynomial , trigonometry , system of equations , algebra
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

2016 German National Olympiad, 1
Tags: algebra , system of equations
Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]

2018 JBMO Shortlist, A4
Tags: algebra , minimum , system of equations , product
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$ Find: a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$ b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.

2021 Serbia Team Selection Test, P3
Tags: diophantine equation , system of equations , number theory , prime number
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

2000 District Olympiad (Hunedoara), 1
Tags: equation , system of equations , number theory , algebra
[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

2024 Ecuador NMO (OMEC), 1
Tags: algebra , system of equations
Find all real solutions: $$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$

1965 IMO, 4
Tags: system of equations , algebra
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

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*}

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: $$x^2 + y^2 + z^2 + w^2 = 4$$ $$\frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2 }+\frac{1}{w^2} = 5 -\frac{1}{(xyzw)^2}$$

2010 Saudi Arabia BMO TST, 4
Tags: system of equations , system , algebra
Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.

IV Soros Olympiad 1997 - 98 (Russia), 11.12
Tags: system of equations , algebra , parameter
Find how many different solutions depending on $a$ has the system of equations : $$\begin{cases} x+z=2a \\ y+u+xz=a-3 \\ yz+xu=2a \\ yu=1 \end{cases}$$

1966 IMO Longlists, 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 $B$?

1968 IMO, 3
Tags: algebra , system of equations , polynomial , discriminant
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

2022 Puerto Rico Team Selection Test, 1
Tags: algebra , system of equations , number theory
Find all triples $(a, b, c)$ of positive integers such that: $$a + b + c = 24$$ $$a^2 + b^2 + c^2 = 210$$ $$abc = 440$$

2004 IberoAmerican, 1
Tags: modular arithmetic , algebra , system of equations , number theory
Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

2007 ITest, 46
Tags: floor function , algebra , system of equations
Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.

1949-56 Chisinau City MO, 56
Tags: algebra , system of equations
Solve the system of equations $$\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}$$

1968 Czech and Slovak Olympiad III A, 1
Tags: algebra , system of equations , inequalities
Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system \begin{align*} x_1x_2 &= a_1, \\ x_2x_3 &= a_2, \\ &\ \vdots \\ x_{n-1}x_n &= a_{n-1}, \\ x_nx_1 &\ge a_n. \end{align*}

1974 Swedish Mathematical Competition, 5
Tags: system of equations , system , algebra
Find the smallest positive real $t$ such that \[\left\{ \begin{array}{l} x_1 + x_3 = 2t x_2 \\ x_2 + x_4 = 2t x_3 \\ x_3 + x_5=2t x_4 \\ \end{array} \right. \] has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.