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

2015 Greece National Olympiad, 3
Tags: geometry
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$. A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$, B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$

2021 New Zealand MO, 8
Tags: combinatorics , board
Two cells in a $20 \times 20$ board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a $20 \times 20$ board such that every cell is adjacent to at most one marked cell?

2024 Korea Winter Program Practice Test, Q1
Tags: geometry
A point $P$ lies inside $\usepackage{gensymb} \angle ABC(<90 \degree)$. Show that there exists a point $Q$ inside $\angle ABC$ satisfying the following condition: [center]For any two points $X$ and $Y$ on the rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ respectively satisfying $\angle XPY = \angle ABC$, it holds that $\usepackage{gensymb} \angle XQY = 180 \degree - 2 \angle ABC.$[/center]

2010 India IMO Training Camp, 6
Tags: floor function , induction , combinatorics
Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

2012 Federal Competition For Advanced Students, Part 1, 2
Tags: modular arithmetic , number theory
Determine all solutions $(n, k)$ of the equation $n!+An = n^k$ with $n, k \in\mathbb{N}$ for $A = 7$ and for $A = 2012$.

1989 National High School Mathematics League, 2
Tags: function
Range of function $f(x)=\arctan x+\frac{1}{2}\arcsin x$ is $\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]$

2025 Taiwan TST Round 2, A
Tags: algebra , functional equation
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and \[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\] for all $x,y\in\mathbb{R}$. [i] Proposed by usjl[/i]

2000 Moldova National Olympiad, Problem 4
Tags: geometry , triangle
The orthocenter $H$ of a triangle $ABC$ is not on the sides of the triangle and the distance $AH$ equals the circumradius of the triangle. Find the measure of $\angle A$.

2003 IMO Shortlist, 6
Tags: inequalities , function , calculus
Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

1998 Greece JBMO TST, 5
Tags: ceiling function , algebra
Let $I$ be an open interval of length $\frac{1}{n}$, where $n$ is a positive integer. Find the maximum possible number of rational numbers of the form $\frac{a}{b}$ where $1 \le b \le n$ that lie in $I$.

2007 Romania Team Selection Test, 2
Tags: parallelogram , trapezoid , geometry
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

CNCM Online Round 1, 2
Tags:
Akshar is reading a $500$ page book, with odd numbered pages on the left, and even numbered pages on the right. Multiple times in the book, the sum of the digits of the two opened pages are $18$. Find the sum of the page numbers of the last time this occurs. Proposed by Minseok Eli Park (wolfpack)

1965 All Russian Mathematical Olympiad, 061
Tags: combinatorics
A society created in the help to the police contains $100$ men exactly. Every evening $3$ men are on duty. Prove that you can not organise duties in such a way, that every couple will meet on duty once exactly.

2012 Pre-Preparation Course Examination, 2
Tags: linear algebra , vector
Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.

1977 Spain Mathematical Olympiad, 2
Tags: group theory , algebra
Prove that all square matrices of the form (with $a, b \in R$), $$\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$$ form a commutative field $K$ when considering the operations of addition and matrix product. Prove also that if $A \in K$ is an element of said field, there exist two matrices of $K$ such that the square of each is equal to $A$.

2006 Estonia National Olympiad, 1
Tags: inequalities
Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real numbers x, for which this value is achieved.

2019 Mediterranean Mathematics Olympiad, 1
Tags: geometry , incircle , inradius
Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

2009 Singapore Senior Math Olympiad, 4
Tags: inequalities , weighted am-gm , logarithm
Given that $ a,b,c, x_1, x_2, ... , x_5 $ are real positives such that $ a+b+c=1 $ and $ x_1.x_2.x_3.x_4.x_5 = 1 $. Prove that \[ (ax_1^2+bx_1+c)(ax_2^2+bx_2+c)...(ax_5^2+bx_5+c)\ge 1\]

2024-IMOC, A3
Tags: sequence , integer sequence
Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying \[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$

2014 Contests, 2
Tags: modular arithmetic , number theory
A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

2005 JHMT, 3
Tags: geometry
Isosceles triangle $ABC$ has angle $\angle BAC = 135^o$ and $AB = 2$. What is its area?

2012 China Team Selection Test, 1
Tags: floor function , function , number theory , logarithm
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that \[\min \{|A|,|B|\}\le\log _2n.\]

2010 Sharygin Geometry Olympiad, 6
Tags: midpoint , square , angle chasing , angle , geometry
Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2016 Saudi Arabia BMO TST, 4
Tags: coloring , combinatorics
Given six three-element subsets of the set $X$ with at least $5$ elements, show that it is possible to color the elements of $X$ in two colors such that none of the given subsets is all in one color.

2022 Caucasus Mathematical Olympiad, 4
Tags: combinatorial geometry , lattice points , combinatorics
Do there exist 2021 points with integer coordinates on the plane such that the pairwise distances between them are pairwise distinct consecutive integers?