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

1999 Korea Junior Math Olympiad, 2
Tags: factorization , sum of cubes , algebra
Three integers are given. $A$ denotes the sum of the integers, $B$ denotes the sum of the square of the integers and $C$ denotes the sum of cubes of the integers(that is, if the three integers are $x, y, z$, then $A=x+y+z$, $B=x^2+y^2+z^2$, $C=x^3+y^3+z^3$). If $9A \geq B+60$ and $C \geq 360$, find $A, B, C$.

2018 BMT Spring, 9
Tags: combinatorics
Let $S$ be the set of integers from $1$ to $13$ inclusive. A permutation of $S$ is a function $f : S \to S$ such that $f(x) \ne f(y)$ if $x \ne y$. For how many distinct permutations $f$ does there exists an $n $ such that $f^n(i) = 13 - i + 1$ for all $i$.

2009 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , number theory
Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$.

2020 Romanian Masters In Mathematics, 4
Tags: function , algebra
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free. [i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]

2013 AMC 10, 8
Tags:
Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline? $ \textbf{(A) }10\qquad\textbf{(B) }16\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40 $

2016 PUMaC Team, 7
Tags: geometry
In triangle $ABC$, let $S$ be on $BC$ and $T$ be on $AC$ so that $AS \perp BC$ and $BT \perp AC$, and let $AS$ and $BT$ intersect at $H$. Let $O$ be the center of the circumcircle of $\vartriangle AHT, P$ be the center of the circumcircle of $\vartriangle BHS$, and $G$ be the other point of intersection (besides $H$) of the two circles. Let $GH$ and $OP$ intersect at $X$. If $AB = 14, BH = 6$, and HA = 11, then $XO - XP$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.

1974 Bulgaria National Olympiad, Problem 1
Tags: combinatorics
Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$-th person is moving $i_n$ places in the right, all people will sit on different places. [i]V. Drenski[/i]

2011 Flanders Math Olympiad, 3
Tags: combinatorics
There are $18$ students in a class. Each student is asked two questions: how many other students have the same first name as you and how many other students have the same surname as you. The answers $0, 1, 2, . . ., 7$ all occur. Prove that there are two students with the same first name and last name.

2004 Postal Coaching, 3
Tags: inequalities , number theory
Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$

2007 Singapore Team Selection Test, 1
Tags: circumcircle , geometry , reflection , geometric transformation , perpendicular bisector
Two circles $ (O_1)$ and $ (O_2)$ touch externally at the point $C$ and internally at the points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $ (O_1)$ and $ (O_2)$ at $C$ meets $(O)$ at $P$ such that $PA=PB$. Prove that $PO$ is perpendicular to $AB$.

1971 IMO Shortlist, 6
Tags: graph coloring , construction , combinatorics
Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

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]

2011 IFYM, Sozopol, 2
Tags: geometry
Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.

PEN E Problems, 6
Tags: algebra , polynomial
Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.

2007 France Team Selection Test, 1
Tags: combinatorics
Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

2022 Indonesia Regional, 5
Tags: mathematical , excalibur , combinatorics
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.

2013 Princeton University Math Competition, 6
Tags: modular arithmetic , trig identities , function , trigonometry , induction , floor function
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.

1955 Miklós Schweitzer, 7
Tags: number theory
[b]7.[/b] Prove that for any odd prime number $p$, the polynomial $2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$ is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b] *This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594

2021 Serbia National Math Olympiad, 4
Tags: geometry
A convex quadrilateral $ABCD$ will be called [i]rude[/i] if there exists a convex quadrilateral $PQRS$ whose points are all in the interior or on the sides of quadrilateral $ABCD$ such that the sum of diagonals of $PQRS$ is larger than the sum of diagonals of $ABCD$. Let $r>0$ be a real number. Let us assume that a convex quadrilateral $ABCD$ is not rude, but every quadrilateral $A'BCD$ such that $A'\neq A$ and $A'A\leq r$ is rude. Find all possible values of the largest angle of $ABCD$.

KoMaL A Problems 2019/2020, A. 774
Tags: geometry , circumcircle , incenter
Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$

2006 Bosnia and Herzegovina Team Selection Test, 5
Tags: perpendicular , geometry , isosceles
Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$. (i) Prove that triangle $PQS$ is isosceles (ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$

1975 Bundeswettbewerb Mathematik, 2
Tags: combinatorial geometry , geometry , polyhedron , 3d geometry , edge
Prove that in each polyhedron there exist two faces with the same number of edges.

1996 Akdeniz University MO, 4
Tags: diameter , point set , geometry
$25$ point in a plane and for all $3$ points, we find $2$ points such that this $2$ points' distance less than $1$ $cm$ . Prove that at least $13$ points in a circle of radius $1$ $cm$.

2006 Harvard-MIT Mathematics Tournament, 2
Tags: geometry
Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle ABP=45^\circ$. Given that $AP=1$ and $CP=2$, compute the area of $ABC$.

Today's calculation of integrals, 881
Tags: calculus computations , trigonometry , integration , definite integral , calculus
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.