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: 451

2017 Saudi Arabia JBMO TST, 2
Tags: equation , algebra
Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions.

1963 IMO Shortlist, 1
Tags: algebra , equation , parametric equation
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

1969 Czech and Slovak Olympiad III A, 1
Tags: rational , algebra , equation , quadratic polynomial
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]

1970 IMO Longlists, 59
Tags: modular arithmetic , digit , equation , number theory
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

2024 Bangladesh Mathematical Olympiad, P3
Tags: algebra , equation
Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?

2013 Poland - Second Round, 4
Tags: algebra , equation
Solve equation $(x^4 + 3y^2)\sqrt{|x + 2| + |y|}=4|xy^2|$ in real numbers $x$, $y$.

1989 IMO Shortlist, 2
Tags: algebra , area , rectangle , equation , geometry
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?

1962 IMO, 4
Tags: trigonometric equation , trigonometry , algebra , equation
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

2006 Grigore Moisil Urziceni, 3
Tags: equation , monotone , algebra , system of equations
Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

1984 IMO Longlists, 13
Tags: diophantine equation , polynomial , equation , quadratic , number theory
Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2019 Danube Mathematical Competition, 1
Tags: equation , diophantine equation , algebra
Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ [i]Lucian Petrescu[/i]

1985 Brazil National Olympiad, 5
Tags: algebra , equation
$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.

1974 Czech and Slovak Olympiad III A, 3
Tags: number theory , digit , sum of digits , permutation , equation , diophantine equation
Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

1966 IMO Longlists, 10
Tags: trigonometric equation , trigonometry , algebra , equation
How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

2020 Middle European Mathematical Olympiad, 4#
Tags: algebra , equation , induction
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that $$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$

1989 IMO Longlists, 4
Tags: calculus , rectangle , equation , area , algebra
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

1968 IMO Shortlist, 22
Tags: number theory , decimal representation , digit , equation
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

1974 IMO Shortlist, 9
Tags: trigonometry , algebra , trigonometric identities , trigonometric substitution , equation
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

2012 Bogdan Stan, 3
Tags: equation , algebra
Find the real numbers $ x,y,z $ that satisfy the following: $ \text{(i)} -2\le x\le y\le z $ $ \text{(ii)} x+y+z=2/3 $ $ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $ [i]Cristinel Mortici[/i]

2017 Latvia Baltic Way TST, 2
Tags: trigonometric equation , algebra , equation , trigonometry
Find all pairs of real numbers $(x, y)$ that satisfy the equation $$\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}$$

2023 China Second Round, 6
Tags: algebra , equation , inequalities
Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$

2023 Bulgaria EGMO TST, 5
Tags: equation , calculate , bound , algebra
The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

1968 IMO Shortlist, 11
Tags: algebra , equation , root
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation \[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]

2005 Germany Team Selection Test, 1
Tags: number theory , equation , divisor
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2004 Germany Team Selection Test, 4
Tags: equation , integer equation
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.