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

2019 Purple Comet Problems, 6
Tags: geometry
A pentagon has four interior angles each equal to $110^o$. Find the degree measure of the fifth interior angle.

2010 Harvard-MIT Mathematics Tournament, 5
Tags: complex number
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^2+y^2$.

2001 AMC 10, 6
Tags: algebra
Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

1986 China Team Selection Test, 4
Tags: induction , geometry
Given a triangle $ABC$ for which $C=90$ degrees, prove that given $n$ points inside it, we can name them $P_1, P_2 , \ldots , P_n$ in some way such that: $\sum^{n-1}_{k=1} \left( P_K P_{k+1} \right)^2 \leq AB^2$ (the sum is over the consecutive square of the segments from $1$ up to $n-1$). [i]Edited by orl.[/i]

2020 Greece JBMO TST, 2
Tags: algebra , inequalities
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that $$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$ When is the equality valid?

1999 Spain Mathematical Olympiad, 3
Tags: combinatorics , board
A one player game is played on the triangular board shown on the picture. A token is placed on each circle. Each token is white on one side and black on the other. Initially, the token at one vertex of the triangle has the black side up, while the others have the white sides up. A move consists of removing a token with the black side up and turning over the adjacent tokens (two tokens are adjacent if they are joined by a segment). Is it possible to remove all the tokens by a sequence of moves? [img]https://cdn.artofproblemsolving.com/attachments/d/2/aabf82a0ddd6907482f27e6e0f1e1b56cd931d.png[/img]

2001 Baltic Way, 15
Tags: algebra , inequalities
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $ Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.

2015 Cuba MO, 5
Tags: algebra , inequalities
Let $a, b$ and $c$ be real numbers such that $0 < a, b, c < 1$. Prove that: $$\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.$$

2024 Nepal Mathematics Olympiad (Pre-TST), Problem 1
Tags: combinatorics , number theory
Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape? [i]( Proposed by Vlad Spǎtaru, Romania)[/i]

2022 Yasinsky Geometry Olympiad, 2
Tags: angle bisector , bisector , angle , geometry
In the triangle $ABC$, angle $C$ is four times smaller than each of the other two angle The altitude $AK$ and the angle bisector $AL$ are drawn from the vertex of the angle $A$. It is known that the length of $AL$ is equal to $\ell$. Find the length of the segment $LK$. (Gryhoriy Filippovskyi)

2018 Mexico National Olympiad, 1
Tags: inequalities , geometry , angle bisector
Let $A$ and $B$ be two points on a line $\ell$, $M$ the midpoint of $AB$, and $X$ a point on segment $AB$ other than $M$. Let $\Omega$ be a semicircle with diameter $AB$. Consider a point $P$ on $\Omega$ and let $\Gamma$ be the circle through $P$ and $X$ that is tangent to $AB$. Let $Q$ be the second intersection point of $\Omega$ and $\Gamma$. The internal angle bisector of $\angle PXQ$ intersects $\Gamma$ at a point $R$. Let $Y$ be a point on $\ell$ such that $RY$ is perpendicular to $\ell$. Show that $MX > XY$

2024 Saint Petersburg Mathematical Olympiad, 2
Tags: combinatorics
$32$ real and $32$ fake coins are given the same appearance. All fake coins weigh equally and less than the real ones, which also all weigh the same. How to determine the type of at least seven coins in six weighings on a scale with two bowls?

2011 China Second Round Olympiad, 3
Tags: floor function , combinatorics
Given $n\ge 4$ real numbers $a_{n}>...>a_{1} > 0$. For $r > 0$, let $f_{n}(r)$ be the number of triples $(i,j,k)$ with $1\leq i<j<k\leq n$ such that $\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r$. Prove that ${f_{n}(r)}<\frac{n^{2}}{4}$.

2013 Saint Petersburg Mathematical Olympiad, 4
Tags: algebra
There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.

2015 HMNT, 7
Tags:
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ meet at $P$. Let the area of triangle $APB$ be $24$ and let the area of triangle $CPD$ be $25$. What is the minimum possible area of quadrilateral $ABCD$?

2016 Japan MO Preliminary, 11
Tags: number theory
How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below? 1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$. 2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.

2016 Korea Winter Program Practice Test, 3
Tags: functional equation , algebra
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

2012 Princeton University Math Competition, A3 / B5
Tags: combinatorics
Jim has two fair $6$-sided dice, one whose faces are labelled from $1$ to $6$, and the second whose faces are labelled from $3$ to $8$. Twice, he randomly picks one of the dice (each die equally likely) and rolls it. Given the sum of the resulting two rolls is $9$, if $\frac{m}{n}$ is the probability he rolled the same die twice where $m, n$ are relatively prime positive integers, then what is $m + n$?

2014 Romania Team Selection Test, 3
Tags: geometry
Let $A_0A_1A_2$ be a scalene triangle. Find the locus of the centres of the equilateral triangles $X_0X_1X_2$ , such that $A_k$ lies on the line $X_{k+1}X_{k+2}$ for each $k=0,1,2$ (with indices taken modulo $3$).

2014 Romania Team Selection Test, 4
Tags: greatest common divisor , function , number theory
Let $n$ be a positive integer and let $A_n$ respectively $B_n$ be the set of nonnegative integers $k<n$ such that the number of distinct prime factors of $\gcd(n,k)$ is even (respectively odd). Show that $|A_n|=|B_n|$ if $n$ is even and $|A_n|>|B_n|$ if $n$ is odd. Example: $A_{10} = \left\{ 0,1,3,7,9 \right\}$, $B_{10} = \left\{ 2,4,5,6,8 \right\}$.

2022 Miklós Schweitzer, 4
Tags:
Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$

2010 Purple Comet Problems, 28
Tags: number theory , relatively prime
There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

2011 Indonesia MO, 6
Tags: modular arithmetic , number theory
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.

2006 AMC 12/AHSME, 8
Tags: arithmetic sequence
How many sets of two or more consecutive positive integers have a sum of 15? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

2022 Taiwan TST Round 3, G
Tags: geometry
Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)