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

PEN H Problems, 81
Tags: diophantine equation , modular arithmetic , number theory , relatively prime
Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.

PEN P Problems, 9
Tags: additive number theory
The integer $9$ can be written as a sum of two consecutive integers: 9=4+5. Moreover it can be written as a sum of (more than one) consecutive positive integers in exactly two ways, namely 9=4+5= 2+3+4. Is there an integer which can be written as a sum of $1990$ consecutive integers and which can be written as a sum of (more than one) consecutive positive integers in exactly $1990$ ways?

2019 Durer Math Competition Finals, 3
Tags: greatest common divisor , relatively prime , totient function , number theory , function
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2004 Gheorghe Vranceanu, 3
Tags: floor function , inequalities , algebra
Let $ a,b,c $ be real numbers satisfying $ \left\lfloor a^2+b^2+c^2 \right\rfloor \le\lfloor ab+bc+ca \rfloor . $ Show that: $$ 2 >\max\left\{ \left| -2a+b+c \right| ,\left| a-2b+c \right| ,\left| a+b-2c \right| \right\} $$ [i]Merticaru[/i]

1985 IMO Longlists, 55
Tags: rhombus , geometry
The points $A,B,C$ are in this order on line $D$, and $AB = 4BC$. Let $M$ be a variable point on the perpendicular to $D$ through $C$. Let $MT_1$ and $MT_2$ be tangents to the circle with center $A$ and radius $AB$. Determine the locus of the orthocenter of the triangle $MT_1T_2.$

2017 AIME Problems, 5
Tags:
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.4
Tags: combinatorics , geometry , combinatorial geometry
There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these $1995 $ segments, it is possible to assemble $664$ acute-angled triangles so that each segment is part of no more than one triangle.

2021 IMO, 3
Tags: geometry , concurrency , circumcenter , moving points
Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.

2021 Novosibirsk Oral Olympiad in Geometry, 3
Tags: geometry , perimeter
In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?

2014 Baltic Way, 14
Tags: circumcircle , geometry
Let $ABCD$ be a convex quadrilateral such that the line $BD$ bisects the angle $ABC.$ The circumcircle of triangle $ABC$ intersects the sides $AD$ and $CD$ in the points $P$ and $Q,$ respectively. The line through $D$ and parallel to $AC$ intersects the lines $BC$ and $BA$ at the points $R$ and $S,$ respectively. Prove that the points $P, Q, R$ and $S$ lie on a common circle.

2006 Tournament of Towns, 7
Tags: combinatorics , geometry , 3d geometry , dodecahedron
An ant craws along a closed route along the edges of a dodecahedron, never going backwards. Each edge of the route is passed exactly twice. Prove that one of the edges is passed both times in the same direction. (Dodecahedron has $12$ faces in the shape of pentagon, $30$ edges and $20$ vertices; each vertex emitting 3 edges). (8)

STEMS 2023 Math Cat A, 7
Tags: probability , algebra , computational , coin toss
Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process ends when one of the players get a $6$, and that player is declared the winner. If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?

2008 ITest, 44
Tags: probability , floor function , arithmetic series , number theory , relatively prime
Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.

2017 Saudi Arabia BMO TST, 1
Tags: product , integer sidelengths , geometry , number theory , right triangle , prime
Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

2017 USAMTS Problems, 3
Tags:
Let $ABC$ be an equilateral triangle with side length $1$. Let $A_1$ and $A_2$ be the trisection points of $AB$ with $A_1$ closer to $A$, $B_1$ and $B_2$ be the trisection points of $BC$ with $B_1$ closer to $B$, and $C_1$ and $C_2$ be the trisection points of $CA$ with $C_1$ closer to $C$. Grogg has an orange equilateral triangle the size of triangle $A_1B_1C_1$. He puts the orange triangle over triangle $A_1B_1C_1$ and then rotates it about its center in the shortest direction until its vertices are over $A_2B_2C_2$. Find the area of the region that the orange triangle traveled over during its rotation.

2010 Indonesia TST, 4
Tags: combinatorics
Prove that the number $ (\underbrace{9999 \dots 99}_{2005}) ^{2009}$ can be obtained by erasing some digits of $ (\underbrace{9999 \dots 99}_{2008}) ^{2009}$ (both in decimal representation). [i]Yudi Satria, Jakarta[/i]

2011 District Olympiad, 4
Tags: algebra , fractional part
Let be a nonzero real number $ a, $ and a natural number $ n. $ Prove the implication: $$ \{ a \} +\left\{\frac{1}{a}\right\} =1 \implies \{ a^n \} +\left\{\frac{1}{a^n}\right\} =1 , $$ where $ \{\} $ is the fractional part.

2023 All-Russian Olympiad Regional Round, 9.5
Tags: geometry
Let $ABCD$ be a cyclic quadrilateral such that the circles with diameters $AB$ and $CD$ touch at $S$. If $M, N$ are the midpoints of $AB, CD$, prove that the perpendicular through $M$ to $MN$ meets $CS$ on the circumcircle of $ABCD$.

2014 Contests, 2
Tags: incenter , circumcircle , reflection , homothety , inversion
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.

2009 Turkey MO (2nd round), 1
Tags: modular arithmetic , number theory
Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.

2020 Durer Math Competition Finals, 3
Tags: number theory , sum of digits
Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?

2010 Iran Team Selection Test, 5
Tags: geometry , circumcircle , geometric transformation , ratio , homothety , radical axis
Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$ \[\angle QXA=\angle QKP\]

2019 ASDAN Math Tournament, 1
Tags: geometry
A square $ABCD$ and point $E$ are drawn in a plane such that lengths $DE < BE$ and $\vartriangle ACE$ is equilateral. Compute $\angle BAE$.

1975 All Soviet Union Mathematical Olympiad, 209
Tags: hexagon , diagonal , area , geometry
Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.

2016 CHMMC (Fall), 5
Tags: 3d geometry , probability , geometry
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.