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

2024 AMC 12/AHSME, 3
Tags:
For how many integer values of $x$ is $|2x|\leq 7\pi?$ $\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

2021 HMNT, 6
Tags: combinatorics
Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of $18$ cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

2016 Purple Comet Problems, 9
Tags:
Find the sum of all perfect squares that divide 2016.

1968 IMO, 1
Tags: trigonometry , algebra , triangle , triangle inequality
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

2017 Harvard-MIT Mathematics Tournament, 2
Tags: perimeter , geometry
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

2018 PUMaC Combinatorics A, 2
Tags: combinatorics , counting , distinguishability
In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.

2024 Middle European Mathematical Olympiad, 5
Tags: circumcircle , geometry
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

2023 USAMTS Problems, 5
Tags: lattice points , number theory
Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that every line in $L$ intersects an odd number of elements of $T$?

2006 Moldova National Olympiad, 11.8
Tags: induction , combinatorics
Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a [i]word[/i]. a) Find the maximal possible length of a [i]word[/i]. b) Find the number of the [i]words[/i] of maximal length.

2000 Manhattan Mathematical Olympiad, 1
Tags:
There are 6 people at a party. Prove that one can [b]either[/b] find a group of $3$ people in which each person is friend with the other two, [b]or[/b] one can find a group of $3$ people in which no two people are friends.

2017 Sharygin Geometry Olympiad, P24
Tags: geometry , tetrahedron , similarity , 3d geometry
Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

2015 Estonia Team Selection Test, 11
Tags: circles , geometry
Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

2017 Greece Team Selection Test, 2
Tags: binomial coefficient , number theory , combinatorics
Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

2017 APMO, 1
Tags: combinatorics
We call a $5$-tuple of integers [i]arrangeable[/i] if its elements can be labeled $a, b, c, d, e$ in some order so that $a-b+c-d+e=29$. Determine all $2017$-tuples of integers $n_1, n_2, . . . , n_{2017}$ such that if we place them in a circle in clockwise order, then any $5$-tuple of numbers in consecutive positions on the circle is arrangeable. [i]Warut Suksompong, Thailand[/i]

2004 USAMTS Problems, 1
Tags:
Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers.

2013 Stanford Mathematics Tournament, 2
Tags:
Consider the numbers $\{24,27,55,64,x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$.

2023 Baltic Way, 6
Tags: combinatorics
Let $n$ be a positive integer. Each cell of an $n \times n$ table is coloured in one of $k$ colours where every colour is used at least once. Two different colours $A$ and $B$ are said to touch each other, if there exists a cell coloured in $A$ sharing a side with a cell coloured in $B$. The table is coloured in such a way that each colour touches at most $2$ other colours. What is the maximal value of $k$ in terms of $n$?

2021 Balkan MO, 3
Tags: number theory
Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite. [i]Proposed by Serbia[/i]

2008 India Regional Mathematical Olympiad, 4
Tags: number theory , divisibility , natural number
Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

2010 IFYM, Sozopol, 6
Tags: geometry
In $\Delta ABC$ $(AB>BC)$ $BM$ and $BL$ $(M,L\in AC)$ are a median and an angle bisector respectively. Let the line through $M$, parallel to $AB$, intersect $BL$ in point $D$ and the line through $L$, parallel to $BC$, intersect $BM$ in point $E$. Prove that $DE\perp BL$.

2022 BAMO, E/3
Tags:
A polygon is called [i]convex[/i] if all its internal angles are smaller than 180$^{\circ}$. Given a convex polygon, prove that one can find three distinct vertices $A$, $P$, and $Q$, where $PQ$ is a side of the polygon, such that the perpendicular from $A$ to the line $PQ$ meets the segment $PQ$ (possible at $P$ of $Q$).

2000 Bundeswettbewerb Mathematik, 3
Tags: geometry
A convex quadrilateral $ABCD$ is inscribed in a semicircle with diameter $AB$. The diagonals $AC,BD$ intersect at $S$, and $T$ is the projection of $S$ on $AB$. Show that $ST$ bisects angle $CTD$.

2014 BMT Spring, 9
Tags: calculus
Two different functions $f, g$ of $x$ are selected from the set of real-valued functions $$\left \{sin x, e^{-x}, x \ln x, \arctan x, \sqrt{x^2 + x} -\sqrt{x^2 + x} -x, \frac{1}{x} \right \}$$ to create a product function $f(x)g(x)$. For how many such products is $\lim_{x\to infty} f(x)g(x)$ finite?

2021 STEMS CS Cat A, Q4
Tags: combinatorics
Let $a_1,a_2, \dots a_n$ be positive real numbers. Define $b_1,b_2, \dots b_n$ as follows. \begin{align*} b_1&=a_1 \\ b_2&=max(a_1,a_2)\\ b_i&=max(b_{i-1},b_{i-2}+a_i) \text{ for } i=3,4 \dots n \end{align*} Also define $c_1,c_2 \dots c_n$ as follows. \begin{align*} c_n&=a_n \\ c_{n-1}&=max(a_n,a_{n-1})\\ c_i&=max(c_{i+1},c_{i+2}+a_i) \text{ for } i=n-2,n-3 \dots 1 \end{align*} Prove that $b_n=c_1$.\\

2016 Sharygin Geometry Olympiad, P11
Tags: geometry , construction , symmedian , circumcircle
Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.