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

2017 AMC 10, 11
Tags: 3d geometry
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

2006 Petru Moroșan-Trident, 1
Tags: polynomial , algebra
Prove that the polynom $ X^3-aX-a+1 $ has three integer roots, for an infinite number of integers $ a. $ [i]Liviu Parsan[/i]

2006 Switzerland Team Selection Test, 1
Tags: inequalities
Let $a,b,c \in \mathbb{R^+}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show $\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}$. :D

2010 Grand Duchy of Lithuania, 3
Tags: combinatorics
At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?

2016 Harvard-MIT Mathematics Tournament, 1
Tags: hmmt , probability
DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.

2016 Harvard-MIT Mathematics Tournament, 6
Tags:
Consider a $2 \times n$ grid of points and a path consisting of $2n-1$ straight line segments connecting all these $2n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called $\textit{efficient}$ if each point is only passed through once and no two line segments intersect. How many efficient paths are there when $n = 2016$?

2021 Polish Junior MO Finals, 1
Tags: positive integer , square root , algebra
Positive integers $a$, $b$ an $n$ satisfy \[ \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. \] Prove that $\sqrt{ab}$ is an integer.

1959 AMC 12/AHSME, 1
Tags: geometry , 3d geometry , percent , percent increase , cube
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is: $ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $

2022 Purple Comet Problems, 9
Tags:
For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.

2008 Czech and Slovak Olympiad III A, 2
Tags: ratio , number theory
At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that $t$ hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is $k>1$. Find the maximal value of $t$.

2022 CMIMC, 1.8
Tags: algebra , number theory
Find the largest $c > 0$ such that for all $n \ge 1$ and $a_1,\dots,a_n, b_1,\dots, b_n > 0$ we have $$\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}$$ [i]Proposed by Grant Yu[/i]

2007 Irish Math Olympiad, 4
Tags: number theory
Find the number of zeros in which the decimal expansion of $ 2007!$ ends. Also find its last non-zero digit.

2005 Austria Beginners' Competition, 2
Tags: number theory , diophantine , inequalities
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .

2000 Finnish National High School Mathematics Competition, 4
Tags: combinatorics
There are seven points on the plane, no three of which are collinear. Every pair of points is connected with a line segment, each of which is either blue or red. Prove that there are at least four monochromatic triangles in the fi gure.

1989 IMO Longlists, 93
Tags: modular arithmetic , number theory , prime number , sequence , power of number
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

1986 IMO Longlists, 40
Tags: function , algebra
Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i \ (1 \leq i \leq m), y_j \ (1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$'s are odd, and $\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.$

2012 Czech-Polish-Slovak Junior Match, 3
Tags: projection , circumcircle , equilateral , fixed , area of a triangle , geometry
Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

2021 Taiwan TST Round 2, N
Tags: number theory
Let $S$ be a set of positive integers such that for every $a,b\in S$, there always exists $c\in S$ such that $c^2$ divides $a(a+b)$. Show that there exists an $a\in S$ such that $a$ divides every element of $S$. [i]Proposed by usjl[/i]

2024 ELMO Shortlist, A4
Tags: algebra
The number $2024$ is written on a blackboard. Each second, if there exist positive integers $a,b,k$ such that $a^k+b^k$ is written on the blackboard, you may write $a^{k'}+b^{k'}$ on the blackboard for any positive integer $k'.$ Find all positive integers that you can eventually write on the blackboard. [i]Srinivas Arun[/i]

1996 Bundeswettbewerb Mathematik, 2
Tags: combinatorics , sum , board
The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

1994 Flanders Math Olympiad, 4
Tags: function , algebra , domain , quadratic
Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.

1976 IMO Longlists, 6
Tags: function , inequalities
For each point $X$ of a given polytope, denote by $f(X)$ the sum of the distances of the point $X$ from all the planes of the faces of the polytope. Prove that if $f$ attains its maximum at an interior point of the polytope, then $f$ is constant.

2018 Middle European Mathematical Olympiad, 6
Tags: circumcircle , geometry
Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$

2025 Polish MO Finals, 2
Tags: number theory , prime number
Positive integers $k, m, n ,p $ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$. Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$.

2002 Italy TST, 2
Tags: floor function , combinatorics
On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.