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

1985 Vietnam Team Selection Test, 2
Tags: trigonometry , function , algebra
Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$

2007 Balkan MO Shortlist, A8
Tags: bmosl , algebra , sequence , recurrence relation
Let $c>2$ and $a_0,a_1, \ldots$ be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any $n$ $\in$ $\mathbb{N}$. Prove, $a_1=0$

2022 JHMT HS, 3
Tags: expected value
Dr. G has a bag of five marbles and enjoys drawing one marble from the bag, uniformly at random, and then putting it back in the bag. How many draws, on average, will it take Dr. G to reach a point where every marble has been drawn at least once?

2007 China Team Selection Test, 2
Tags: combinatorics
Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

1952 AMC 12/AHSME, 10
Tags:
An automobile went up a hill at a speed of $ 10$ miles an hour and down the same distance at a speed of $ 20$ miles an hour. The average speed for the round trip was: $ \textbf{(A)}\ 12\frac {1}{2} \text{ mph} \qquad\textbf{(B)}\ 13\frac {1}{3} \text{ mph} \qquad\textbf{(C)}\ 14\frac {1}{2} \text{ mph} \qquad\textbf{(D)}\ 15 \text{ mph}$ $ \textbf{(E)}\ \text{none of these}$

2019 Germany Team Selection Test, 1
Tags: number theory , arithmetic function , number of divisors , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2010 Princeton University Math Competition, 6
Tags:
Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.

1966 Putnam, B2
Tags:
Prove that among any ten consecutive integers at least one is relatively prime to each of the others.

1999 Turkey MO (2nd round), 3
Tags: function , combinatorics
For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions $f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$ such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.

2023 Malaysian IMO Training Camp, 3
Tags: geometry
Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$. [i]Proposed by Wong Jer Ren[/i]

2020 Jozsef Wildt International Math Competition, W5
Tags: real analysis , sequence
Let $(a_n)_{n\ge1}$ and $(b_n)_{n\ge1}$ be positive real sequences such that $$\lim_{n\to\infty}\frac{a_{n+1}-a_n}n=a\in\mathbb R^*_+\enspace\text{and}\enspace\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in\mathbb R^*_+$$ Compute $$\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

2012 HMNT, 2
Tags: number theory
Find the number of ordered triples of divisors $(d_1, d_2, d_3)$ of $360$ such that $d_1d_2d_3$ is also a divisor of $360$. In this section, the word [i]divisor [/i]is used to refer to a [i]positive divisor[/i] of an integer.

2010 Balkan MO, 1
Tags: rearrangement inequality , inequalities
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2024 HMNT, 10
Tags:
Isabella the geologist discovers a diamond deep underground via an X-ray machine. The diamond has the shape of a convex cyclic pentagon $PABCD$ with $AD|| BC$. Soon after the discovery, her X-ray breaks, and she only recovers partial information about its dimensions. She knows that $AD = 70, BC = 55, PA : PD = 3 : 4$, and $PB : PC = 5 : 6$. Compute $PB$.

2018-IMOC, C3
Tags: combinatorics
Given an $a\times b$ chessboard where $a,b\ge3$, alice wants to use only $L$-dominoes (as the figure shows) to cover this chessboard. How many grids, at least, are covered even times? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi82LzhmZDkwMGQzZjU3M2QxMzk4Y2NjNDg5ZTMwM2ZmYjJiMWU3MmUwLnBuZw==&rn=MjAxOC1DMy5wbmc=[/img]

2015 NIMO Summer Contest, 10
Tags: geometry , circumference , incenter
Let $ABCD$ be a tetrahedron with $AB=CD=1300$, $BC=AD=1400$, and $CA=BD=1500$. Let $O$ and $I$ be the centers of the circumscribed sphere and inscribed sphere of $ABCD$, respectively. Compute the smallest integer greater than the length of $OI$. [i] Proposed by Michael Ren [/i]

1990 Tournament Of Towns, (268) 2
Tags: geometry , locus , semicircle
A semicircle $S$ is drawn on $AB$ as diameter. For an arbitrary point $C$ in $S$ ($C\ne A$,$ C \ne B$), squares are attached to sides $AC$ and $BC$ of triangle $ABC$ outside the triangle. Find the locus of the midpoint of the segment joining the centres of the squares as $C$ moves along $S$. (J Tabov, Sofia)

2016 BMT Spring, 11
Tags: algebra
The roots of the polynomial $x^3 - \frac32 x^2 - \frac14 x + \frac38 = 0$ are in arithmetic progression. What are they?

2020-2021 Fall SDPC, 5
Tags: geometry
Let $ABC$ be a triangle with area $1$. Let $D$ be a point on segment $BC$. Let points $E$ and $F$ on $AC$ and $AB$, respectively, satisfy $DE || AB$ and $DF || AC$. Compute, with proof, the area of the quadrilateral with vertices at $E$, $F$, the midpoint of $BD$, and the midpoint of $CD$.

2008 Irish Math Olympiad, 3
Tags: number theory
Find $ a_3,a_4,...,a{}_2{}_0{}_0{}_8$, such that $ a_i =\pm1$ for $ i=3,...,2008$ and $ \sum\limits_{i=3}^{2008} a_i2^i = 2008$ and show that the numbers $ a_3,a_4,...,a_{2008}$ are uniquely determined by these conditions.

1996 Romania National Olympiad, 3
Tags: geometry , rotation , regular polygon
Let $P$ a convex regular polygon with $n$ sides, having the center $O$ and $\angle xOy$ an angle of measure $a$, $a \in (0,k)$. Let $S$ be the area of the common part of the interiors of the polygon and the angle. Find, as a function of $n$, the values of $a$ such that $S$ remains constant when $\angle xOy$ is rotating around $O$.

2023 Kyiv City MO Round 1, Problem 5
Tags: combinatorics , graph theory
In a galaxy far, far away there are $225$ inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The [i]influence[/i] of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions? [i]Proposed by Arsenii Nikolaev, Bogdan Rublov[/i]

1995 IMO Shortlist, 3
Tags: trigonometry , harmonic division , power of a point , incircle , geometry
The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively. Show that $ E, F, Z, Y$ are concyclic.

2007 Mexico National Olympiad, 2
Tags: combinatorics
In each square of a $6\times6$ grid there is a lightning bug on or off. One move is to choose three consecutive squares, either horizontal or vertical, and change the lightning bugs in those $3$ squares from off to on or from on to off. Show if at the beginning there is one lighting bug on and the rest of them off, it is not possible to make some moves so that at the end they are all turned off.

2012 IFYM, Sozopol, 3
Tags: geometry , regular polygon
In a circle with radius 1 a regular n-gon $A_1 A_2...A_n$ is inscribed. Calculate the product: $A_1 A_2.A_1 A_3 \dots A_1 A_{n-1} .A_1 A_n$.