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

1989 Putnam, A3

Tags: search , algebra
Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).

Cono Sur Shortlist - geometry, 1993.8

In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.

2015 NIMO Summer Contest, 11

Tags: prime , square
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers? [i] Proposed by Lewis Chen [/i]

2017 Estonia Team Selection Test, 12

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

2022 Romania EGMO TST, P1

Tags: algebra , function
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]

2023 Sharygin Geometry Olympiad, 8.8

Tags: geometry
Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.

2014 PUMaC Team, 8

Tags:
Let $n^2-6n+1=0$. Find $n^6+\frac1{n^6}$.

2013 Online Math Open Problems, 4

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For how many ordered pairs of positive integers $(a,b)$ with $a,b<1000$ is it true that $a$ times $b$ is equal to $b^2$ divided by $a$? For example, $3$ times $9$ is equal to $9^2$ divided by $3$. [i]Ray Li[/i]

2011 Tournament of Towns, 6

Tags: algebra
A car goes along a straight highway at the speed of $60$ km per hour. A $100$ meter long fence is standing parallel to the highway. Every second, the passenger of the car measures the angle of vision of the fence. Prove that the sum of all angles measured by him is less than $1100$ degrees.

2014 ELMO Shortlist, 6

Show that the numerator of \[ \frac{2^{p-1}}{p+1} - \left(\sum_{k = 0}^{p-1}\frac{\binom{p-1}{k}}{(1-kp)^2}\right) \] is a multiple of $p^3$ for any odd prime $p$. [i]Proposed by Yang Liu[/i]

1968 Miklós Schweitzer, 7

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2014 Romania National Olympiad, 2

Tags: algebra
Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then [b]a)[/b] $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $ [b]b)[/b] $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $ Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.

2016 AMC 10, 14

Tags:
How many ways are there to write $2016$ as the sum of twos and threes, ignoring order? (For example, $1008\cdot 2 + 0\cdot 3$ and $402\cdot 2 + 404\cdot 3$ are two such ways.) $\textbf{(A)}\ 236\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 337\qquad\textbf{(D)}\ 403\qquad\textbf{(E)}\ 672$

1988 IberoAmerican, 2

Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$. Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$, then $p=a+c$.

2017 Dutch BxMO TST, 3

Tags: geometry
Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

2009 Kyrgyzstan National Olympiad, 5

Prove for all natural $n$ that $\left. {{{40}^n} \cdot n!} \right|(5n)!$

2019 Dutch BxMO TST, 4

Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?

1994 India National Olympiad, 2

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.

2007 JBMO Shortlist, 2

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

LMT Speed Rounds, 3

Tags: alg , speed
Sam Wang decides to evaluate an expression of the form $x +2 \cdot 2+ y$. However, he unfortunately reads each ’plus’ as a ’times’ and reads each ’times’ as a ’plus’. Surprisingly, he still gets the problem correct. Find $x + y$. [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{4}$ We have $x+2*2+y=x \cdot 2+2 \cdot y$. When simplifying, we have $x+y+4=2x+2y$, and $x+y=4$. [/hide]

2009 Today's Calculation Of Integral, 485

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

LMT Team Rounds 2010-20, A1 B9

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Ben writes the string $$\underbrace{111\ldots 11}_{2020 \text{ digits}}$$on a blank piece of paper. Next, in between every two consecutive digits, he inserts either a plus sign $(+)$ or a multiplication sign $(\times)$. He then computes the expression using standard order of operations. Find the number of possible distinct values that Ben could have as a result. [i]Proposed by Taiki Aiba[/i]

2015 HMNT, 8

Tags:
Find $ \textbf{any} $ quadruple of positive integers $(a,b,c,d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.

2018 HMNT, 8

Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?

2017 AMC 8, 15

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In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture. [asy] fill((0.5, 4.5)--(1.5,4.5)--(1.5,2.5)--(0.5,2.5)--cycle,lightgray); fill((1.5,3.5)--(2.5,3.5)--(2.5,1.5)--(1.5,1.5)--cycle,lightgray); label("$8$", (1, 0)); label("$C$", (2, 0)); label("$8$", (3, 0)); label("$8$", (0, 1)); label("$C$", (1, 1)); label("$M$", (2, 1)); label("$C$", (3, 1)); label("$8$", (4, 1)); label("$C$", (0, 2)); label("$M$", (1, 2)); label("$A$", (2, 2)); label("$M$", (3, 2)); label("$C$", (4, 2)); label("$8$", (0, 3)); label("$C$", (1, 3)); label("$M$", (2, 3)); label("$C$", (3, 3)); label("$8$", (4, 3)); label("$8$", (1, 4)); label("$C$", (2, 4)); label("$8$", (3, 4));[/asy] $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }24\qquad\textbf{(E) }36$