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

Russian TST 2020, P1
Tags: algebra , functional equation
Determine all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ satisfying $xf(xf(2y))=y+xyf(x)$ for all $x,y>0$.

2007 Kazakhstan National Olympiad, 4
Tags: function , induction , algebra
Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition $f (xf (y) + f (x)) = 2f (x) + xy$ for any real $x$ and $y$.

2022 JHMT HS, 7
Tags: inequalities , algebra
Find the least positive integer $N$ such that there exist positive real numbers $a_1,a_2,\dots,a_N$ such that \[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]

1998 Harvard-MIT Mathematics Tournament, 6
Tags: sfft , symmetry
How many pairs of positive integers $(a,b)$ with $a\leq b$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}$?

2020 Brazil Cono Sur TST, 2
Tags: geometry , parallelogram
Let $ABC$ be a triangle, the point $E$ is in the segment $AC$, the point $F$ is in the segment $AB$ and $P=BE\cap CF$. Let $D$ be a point such that $AEDF$ is a parallelogram, Prove that $D$ is in the side $BC$, if and only if, the triangle $BPC$ and the quadrilateral $AEPF$ have the same area.

2003 APMO, 1
Tags: polynomial , algebra
Let $a,b,c,d,e,f$ be real numbers such that the polynomial \[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \] factorises into eight linear factors $x-x_i$, with $x_i>0$ for $i=1,2,\ldots,8$. Determine all possible values of $f$.

2022 EGMO, 5
Tags: combi , combinatorics
For all positive integers $n$, $k$, let $f(n, 2k)$ be the number of ways an $n \times 2k$ board can be fully covered by $nk$ dominoes of size $2 \times 1$. (For example, $f(2, 2)=2$ and $f(3, 2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2k)$ is odd.

2023 LMT Fall, 10
Tags: combinatorics
Aidan and Andrew independently select distinct cells in a $4 $ by $4$ grid, as well as a direction (either up, down, left, or right), both at random. Every second, each of them will travel $1$ cell in their chosen direction. Find the probability that Aidan and Andrew willmeet (be in the same cell at the same time) before either one of them hits an edge of the grid. (If Aidan and Andrew cross paths by switching cells, it doesn’t count as meeting.)

2015 Chile National Olympiad, 6
Tags: combinatorics , coloring
On the plane, a closed curve with simple auto intersections is drawn continuously. In the plane a finite number is determined in this way from disjoint regions. Show that each of these regions can be completely painted either white or blue, so that every two regions that share a curve segment at its edges, they always have different colors. Clarification: a car intersection is simple if looking at a very small disk around from her, the curve looks like a junction $\times$.

2021 Kyiv Mathematical Festival, 1
Tags: arithmetic sequence , combinatorial geometry , geometry
Is it possible to mark four points on the plane so that the distances between any point and three other points form an arithmetic progression? (V. Brayman)

2011 Belarus Team Selection Test, 2
Tags: algebra , inequalities
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$ I.Voronovich

2008 Tournament Of Towns, 3
Tags: perimeter , geometry
Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$

2014 South East Mathematical Olympiad, 7
Tags: exterior angle , geometry
Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.

2021 Romania EGMO TST, P3
Tags: combinatorics
Determine all pairs of positive integers $(m,n)$ for which an $m\times n$ rectangle can be tiled with (possibly rotated) L-shaped trominos.

2018 Online Math Open Problems, 24
Tags:
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that [list] [*] $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$, and [*] $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$. [/list] Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.) [i]Proposed by Ankan Bhattacharya[/i]

2010 Indonesia TST, 3
Tags: number theory
Let $ x$, $ y$, and $ z$ be integers satisfying the equation \[ \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}.\] Determine the greatest value that $ z$ can take. [i]Budi Surodjo, Jogjakarta[/i]

2019 ELMO Shortlist, N4
Tags: number theory , divisibility
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. [i]Proposed by Carl Schildkraut and Holden Mui[/i]

2023 Turkey Junior National Olympiad, 3
Tags: number theory , divisibility , relatively prime
Let $m,n$ be relatively prime positive integers. Prove that the numbers $$\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}$$ cannot be integer at the same time.

2008 Harvard-MIT Mathematics Tournament, 1
Tags: quadratic
Determine all pairs $ (a,b)$ of real numbers such that $ 10, a, b, ab$ is an arithmetic progression.

2021 OMMock - Mexico National Olympiad Mock Exam, 5
Tags: combinatorics
Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that: [list] [*] No two numbers are equal. [*] If a number $m$ is placed on square $C$, then at least $k$ of the squares orthogonally adjacent to $C$ have a multiple of $m$ written on them. [/list] What is the greatest value of $k$ for which this is possible?

2005 Slovenia Team Selection Test, 2
Tags: functional , functional equation , algebra
Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

2017 BMT Spring, 16
Tags: geometry , geometric inequality , area of a triangle , centroid , area
Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.

2020 Harvard-MIT Mathematics Tournament, 7
Tags:
Find the sum of all positive integers $n$ for which \[\frac{15\cdot n!^2+1}{2n-3}\] is an integer. [i]Proposed by Andrew Gu.[/i]

2018 Turkey Team Selection Test, 9
Tags: simson line , geometry
For a triangle $T$ and a line $d$, if the feet of perpendicular lines from a point in the plane to the edges of $T$ all lie on $d$, say $d$ focuses $T$. If the set of lines focusing $T_1$ and the set of lines focusing $T_2$ are the same, say $T_1$ and $T_2$ are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.

2015 India PRMO, 1
Tags: arithmetic
$1.$ A man walks a certain distance and rides back in $3\frac{3}{4};$ he could ride both ways in $2\frac{1}{2}$ hours. How many hours would it take him to walk both ways $?$