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 Harvard-MIT Mathematics Tournament, 7
Tags: calculus , geometry
A line in the plane is called [i]strange[/i] if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called [i]charming[/i] if it lies in the first quadrant and also lies [b]below[/b] some strange line. What is the area of the set of all charming points?

2022 IFYM, Sozopol, 7
Tags: geometry , concyclic
Given an acute-angled $\vartriangle ABC$ with orthocenter $H$ and altitude $CC_1$. Points $D, E$ and $F$ lie on the segments $AC$, $BC$ and $AB$ respectively, so that $DE \parallel AB$ and $EF \parallel AC$. Denote by $Q$ the symmetric point of $H$ wrt to the midpoint of $DE$. Let $BD \cap CF = P$. If $HP \parallel AB$, prove that the points $C_1, D, Q$ and $E$ lie on a circle.

2024 ITAMO, 3
Tags: egyptian fractions , reciprocal sum , number theory
A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\] (a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

2020 CHMMC Winter (2020-21), 7
Tags: combinatorics
Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that: 1. There is an odd number of red segments connected to $A$} 2. The number of red segments connected to each of the other points are all different Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.

1976 Czech and Slovak Olympiad III A, 3
Tags: construction , geometry , trapezoid , geometric construction
Consider a half-plane with the boundary line $p$ and two points $M,N$ in it such that the distances $Mp$ and $Np$ are different. Construct a trapezoid $MNPQ$ with area $MN^2$ such that $P,Q\in p.$ Discuss conditions of solvability.

2010 Postal Coaching, 6
Tags: number theory , algebra , polynomial , relatively prime , modular arithmetic , induction
Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

2001 Greece Junior Math Olympiad, 3
Tags: combinatorics
We are given $8$ different weights and a balance without a scale. (a) Find the smallest number of weighings necessary to find the heaviest weight. (b) How many weighting is further necessary to find the second heaviest weight?

2004 National High School Mathematics League, 2
Tags:
In rectangular coordinate system, define two sequences of points: $(A_n)$ on the positive half of the $y$-axis and $(B_n)$ on the curve $y=\sqrt{2x}(x\geq0)$ satisfy that $|OA_n|=|OB_n|=\frac{1}{n}$. $a_n$ is the $x$-intercept of line $A_nB_n$, and the $x$-axis of $B_n$ is $b_n$, $n\in\mathbb{Z}_+$. Prove: [b](a)[/b] $a_n>a_{n+1}>4,n\in\mathbb{Z}_+$; [b](b)[/b] There exists $n_0\in\mathbb{Z}_+$, such that $\forall n>n_0$, $\frac{b_2}{b_1}+\frac{b_3}{b_2}+\cdots +\frac{b_n}{b_{n-1}}+\frac{b_{n+1}}{b_n}<n-2004$.

2000 Estonia National Olympiad, 1
Tags: algebra , arithmetic sequence , geometric sequence , arithmetic progression
Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence. Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.

2009 Greece JBMO TST, 1
Tags: sum , algebra , number theory
One pupil has $7$ cards of paper. He takes a few of them and tears each in $7$ pieces. Then, he choses a few of the pieces of paper that he has and tears it again in $7$ pieces. He continues the same procedure many times with the pieces he has every time. Will he be able to have sometime $2009$ pieces of paper?

1967 IMO Shortlist, 6
Tags: geometry , point set , sequence
Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$

1991 Arnold's Trivium, 6
Tags: function
In the $(x,y)$-plane sketch the curve given parametrically by $x=2t-4t^3$, $y=t^2-3t^4$.

2007 Peru IMO TST, 4
Tags: inequalities
Let $a,b$ and $c$ be sides of a triangle. Prove that: $\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$

2016-2017 SDML (Middle School), 10
Tags: inequalities
For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$ $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

2022 China Second Round A1, 4
Tags: combinatorics
Given $r\in\mathbb{R}$. Alice and Bob plays the following game: An equation with blank is written on the blackboard as below: $$S=|\Box-\Box|+|\Box-\Box|+|\Box-\Box|$$ Every round, Alice choose a real number from $[0,1]$ (not necessary to be different from the numbers chosen before) and Bob fill it in an empty box. After 6 rounds, every blank is filled and $S$ is determined at the same time. If $S\ge r$ then Alice wins, otherwise Bob wins. Find all $r$ such that Alice can guarantee her victory.

1969 All Soviet Union Mathematical Olympiad, 116
Tags: algebra
There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.

2008 Greece JBMO TST, 2
Tags: inequality , algebra
If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

2022 Grosman Mathematical Olympiad, P1
Tags: number theory , factorial
For each positive integer $n$ denote: \[n!=1\cdot 2\cdot 3\dots n\] Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.

2025 Balkan MO, 1
Tags: number theory
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that: $(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$; $(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$. Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

2007 Germany Team Selection Test, 2
Tags: homothety , geometry , circumcircle , trapezoid , ratio
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

2017 AMC 12/AHSME, 12
Tags: complex number
What is the sum of the roots of $z^{12} = 64$ that have a positive real part? $\textbf{(A) }2 \qquad\textbf{(B) }4 \qquad\textbf{(C) }\sqrt{2} +2\sqrt{3}\qquad\textbf{(D) }2\sqrt{2}+ \sqrt{6} \qquad\textbf{(E) }(1 + \sqrt{3}) + (1+\sqrt{3})i$

2006 MOP Homework, 4
Tags: combinatorics , game , winning strategy , game strategy
For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.

2013 NIMO Problems, 6
Tags: triangle inequality , inequalities
Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$. It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$. [i]Proposed by Evan Chen[/i]

2024 Korea Junior Math Olympiad, 4
Tags: number theory
find all positive integer n such that there exists positive integers (a,b) such that (a^n + b^n)/n! is a positive integer smaller than 101

2014 BMT Spring, 14
Tags: algebra
Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0$. Find the sum of all possible values of $x$.