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.

AND:
OR:
NO:

Found problems: 85335

2021 Tuymaada Olympiad, 1
Tags: inequalities
Polynomials $F$ and $G$ satisfy: $$F(F(x))>G(F(x))>G(G(x))$$ for all real $x$.Prove that $F(x)>G(x)$ for all real $x$.

2017 Istmo Centroamericano MO, 1
Tags: geometry , angle bisector
Let $ABC$ be a triangle with $\angle ABC = 90^o$ and $AB> BC$. Let $D$ be a point on side $AB$ such that $BD = BC$. Let $E$ be the foot of the perpendicular from $D$ on $AC$, and $F$ the reflection of $B$ wrt $CD$. Show that $EC$ is the bisector of angle $\angle BEF$.

1987 AMC 12/AHSME, 27
Tags: geometry , 3d geometry
A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y$, $y=z$ and $z=x$. How many pieces are there? (No cheese is moved until all three cuts are made.) $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 9 $

2016 India Regional Mathematical Olympiad, 1
Tags: geometry
Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $ \Gamma $ be its circumcircle and let $O$ be the centre of $ \Gamma $ . let $CO$ meet $ \Gamma$ in $D .$ Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at $E.$ Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.

2001 Moldova National Olympiad, Problem 3
Tags: game , combinatorics
During a fight, each of the $2001$ roosters has torn out exactly one feather of another rooster, and each rooster has lost a feather. It turned out that among any three roosters there is one who hasn’t torn out a feather from any of the other two roosters. Find the smallest $k$ with the following property: It is always possible to kill $k$ roosters and place the rest into two henhouses in such a way that no two roosters, one of which has torn out a feather from the other one, stay in the same henhouse.

Durer Math Competition CD 1st Round - geometry, 2014.C4
Tags: geometry , pentagon , area
$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?

2016 China Team Selection Test, 3
Tags: geometry , cyclic quadrilateral , incenter
In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.

2021-IMOC, A2
Tags: algebra
For any positive integers $n$, find all $n$-tuples of complex numbers $(a_1,..., a_n)$ satisfying $$(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.$$ Proposed by USJL.

2009 Putnam, B4
Tags: algebra , polynomial , vector , calculus , integration , complex analysis
Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

2009 Harvard-MIT Mathematics Tournament, 9
Tags: function
How many functions $f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}$ satisfy $f(f(x)) = f(x)$ for all $x\in\{ 1,2, 3, 4, 5\}$?

2004 Estonia National Olympiad, 5
Tags: alphabet , combinatorics
The alphabet of language $BAU$ consists of letters $B, A$, and $U$. Independently of the choice of the $BAU$ word of length n from which to start, one can construct all the $BAU$ words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: $BA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU$ or $AA \to UB$. Given that $BBAUABAUUABAUUUABAUUUUABB$ is a $BAU$ word, does $BAU$ have a) the word $BUABUABUABUABAUBAUBAUBAUB$ ? b) the word $ABUABUABUABUAUBAUBAUBAUBA$ ?

2023 Stanford Mathematics Tournament, 2
Tags:
Every cell in a $5\times5$ grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree? [center][img]https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png[/img][/center]

MathLinks Contest 6th, 6.1
Tags: algebra , inequalities
Let $p > 1$ and let $a, b, c, d$ be positive numbers such that $$(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.$$ Find all values of the ratio $ R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}$ (depending on the parameter $p$)

1977 Germany Team Selection Test, 1
Tags: rearrangement inequality , inequality , permutation , algebra
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3.1
Tags: geometry , equal segments , pentagon , regular pentagon
Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.

2015 Greece Team Selection Test, 1
Tags: number theory
Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$

1983 National High School Mathematics League, 7
Tags:
$P$ is a point on the plane which square $ABCD$ belongs to, satisfying that $\triangle PAB,\triangle PBC,\triangle PCD,\triangle PDA$ are isosceles triangles. What's the number of such points? $\text{(A)}9\qquad\text{(B)}17\qquad\text{(C)}1\qquad\text{(D)}5$

2024 Korea Junior Math Olympiad (First Round), 12.
Tags: algebra , maximum value
For reals $x,y$, find the maximum of A. $ A=\frac{-x^2-y^2-2xy+30x+30y+75}{3x^2-12xy+12y^2+12} $

PEN R Problems, 6
Tags: the geometry of numbers , geometry
Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

2014 Online Math Open Problems, 8
Tags:
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying \begin{align*} 2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\ 2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\ 2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\ 2a_4+a_5 &= 6 + a_1 \end{align*} Compute $a_1+a_2+a_3+a_4+a_5$. [i]Proposed by Evan Chen[/i]

2019 Istmo Centroamericano MO, 3
Tags: equal angles , tangent circle , geometry
Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.

1997 All-Russian Olympiad, 3
Tags: geometry , rectangle , combinatorics
The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. [i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]

2006 CHKMO, 1
Tags: pigeonhole principle , induction , combinatorics
On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

2023 China Girls Math Olympiad, 2
Tags: combinatorics
On an $8\times 8$ chessboard, place a stick on each edge of each grid (on a common edge of two grid only one stick will be placed). What is the minimum number of sticks to be deleted so that the remaining sticks do not form any rectangle?

1999 Mexico National Olympiad, 2
Tags: arithmetic sequence , number theory , prime
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.