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: 85335

2009 USA Team Selection Test, 9
Tags: inequalities , search , function , calculus , derivative , algebra , polynomial
Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]

2018 Pan African, 1
Tags:
Find all functions $f : \mathbb Z \to \mathbb Z$ such that $$(f(x + y))^2 = f(x^2) + f(y^2)$$ for all $x, y \in \mathbb Z$.

2005 Tuymaada Olympiad, 1
Tags: combinatorics
The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers? [i]Proposed by S. Berlov[/i]

2019 India PRMO, 29
Tags: geometry
Let $ABC$ be an acute angled triangle with $AB=15$ and $BC=8$. Let $D$ be a point on $AB$ such that $BD=BC$. Consider points $E$ on $AC$ such that $\angle DEB=\angle BEC$. If $\alpha$ denotes the product of all possible values of $AE$, find $\lfloor \alpha \rfloor$ the integer part of $\alpha$.

1996 Tournament Of Towns, (504) 1
Tags: number theory , consecutive , algebra , sum of squares
Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers? (Inspired by a diagram in an old text book)

2010 Contests, 2
Tags: function , inequalities , cauchy inequality , absolute value , algebra
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

1940 Moscow Mathematical Olympiad, 067
Tags: compare , algebra , factorial
Which is greater: $300!$ or $100^{300}$?

2013 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: algebra , inequalities
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$

2000 German National Olympiad, 5
Tags: coloring , combinatorics , combinatorial geometry
(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors. (b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.

1990 Hungary-Israel Binational, 3
Tags: induction , algebra
Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]

2000 AMC 10, 15
Tags:
Two non-zero real numbers, $a$ and $b$, satisfy $ab=a-b$. Which of the following is a possible value of $\frac ab+\frac ba-ab$? $\text{(A)}\ -2\qquad\text{(B)}\ -\frac12\qquad\text{(C)}\ \frac13\qquad\text{(D)}\ \frac12\qquad\text{(E)}\ 2$

2022 Yasinsky Geometry Olympiad, 2
Tags: perimeter , geometry
In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$. (Gryhoriy Filippovskyi)

2017 HMNT, 5
Tags: algebra
Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $| a + b\omega + c\omega^2|$.

2023 Moldova EGMO TST, 4
Tags: number theory , prime number , algebra , system of equations
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

2025 Caucasus Mathematical Olympiad, 2
Tags: combinatorics
There are $30$ children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.

2002 Federal Competition For Advanced Students, Part 2, 2
Tags: number theory
Let $b$ be a positive integer. Find all $2002$−tuples $(a_1, a_2,\ldots , a_{2002})$, of natural numbers such that \[\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.\]

2015 Indonesia MO Shortlist, A4
Tags: algebra , functional equation , function
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that \[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$ and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.

2018 Peru Iberoamerican Team Selection Test, P1
Tags: algebra
Let $p, q$ be real numbers. Knowing that there are positive real numbers $a, b, c$, different two by two, such that $$p=\frac{a^2}{(b-c)^2}+\frac{b^2}{(a-c)^2}+\frac{c^2}{(a-b)^2},$$ $$q=\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}+\frac{1}{(b-a)^2}$$ calculate the value of $$\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}$$ in terms of $p, q$.

MOAA Team Rounds, 2022.1
Tags: combinatorics
Consider the $5$ by $5$ equilateral triangular grid as shown: [img]https://cdn.artofproblemsolving.com/attachments/1/2/cac43ae24fd4464682a7992e62c99af4acaf8f.png[/img] How many equilateral triangles are there with sides along the gridlines?

2010 Romania Team Selection Test, 1
Tags: parallelogram , geometric transformation , homothety , rectangle , geometry
Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point. [i]Mihai Chis[/i]

2005 Gheorghe Vranceanu, 4
Tags: geometry
Let be a triangle $ ABC $ and the points $ E,F,M,N $ positioned in this way: $ E,F $ on the segment $ BC $ (excluding its endpoints), $ M $ on the segment $ AC $ (excluding its endpoints) and $ N $ on the segment $ AC $ (excluding its endpoints). Knowing that $ BAE $ is similar to $ FAC $ and that $ BE=BM,FC=CN,AM=AN, $ show that $ ABC $ is isosceles.

1993 AIME Problems, 10
Tags:
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?

2001 Miklós Schweitzer, 3
Tags: abstract algebra , ring theory , matrices
How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?

1997 Flanders Math Olympiad, 4
Tags: geometry , geometric transformation , reflection , algebra
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)

2014 IFYM, Sozopol, 5
Tags: geometry
Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.