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

2009 Sharygin Geometry Olympiad, 11
Tags: circumcircle , least common multiple , trapezoid , geometry , number theory
Given quadrilateral $ ABCD$. The circumcircle of $ ABC$ is tangent to side $ CD$, and the circumcircle of $ ACD$ is tangent to side $ AB$. Prove that the length of diagonal $ AC$ is less than the distance between the midpoints of $ AB$ and $ CD$.

2019 Saudi Arabia JBMO TST, 4
Tags: combinatorics
Given is a grid 11x11 with 121 cells. Four of them are colored in black, the rest are white. We have to cut a completely white rectangle (it could be a square and the rectangle must have its sides parralel to the lines of the grid), so that this rectangle has maximal possible area. What largest area of this rectangle we can guarantee? (We can cut this rectangle for every placement of the black squares)

Novosibirsk Oral Geo Oly IX, 2021.4
Tags: geometry , square , tangent circle
A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

2014 India IMO Training Camp, 3
Tags: additive combinatorics , sequence , combinatorics , function
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

2011 ELMO Shortlist, 7
Tags: graph theory , combinatorics
Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges. [i]David Yang.[/i]

1998 Harvard-MIT Mathematics Tournament, 9
Tags: algebra , function , rational function
Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$.

1974 AMC 12/AHSME, 6
Tags:
For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$; then $ \textbf{(A)}\ \text{"*" is commutative but not associative} \\ \qquad\textbf{(B)}\ \text{"*" is associative but not commutative} \\ \qquad\textbf{(C)}\ \text{"*" is neither commutative nor associative} \\ \qquad\textbf{(D)}\ \text{"*" is commutative and associative} \\ \qquad\textbf{(E)}\ \text{none of these} $

2021 JHMT HS, 7
Tags: general , probability
At a prom, there are $4$ boys and $3$ girls. Each boy picks a girl to dance with, and each girl picks a boy to dance with. Assuming that each choice is uniformly random, the probability that at least one boy and one girl choose each other as dance partners is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Compute $p+q.$

2023 Korea Junior Math Olympiad, 4
Tags: combinatorics
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player". [b](Condition)[/b] For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$. It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.

2019 IMO Shortlist, G4
Tags: triangle , geometry
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

1996 Romania National Olympiad, 3
Tags: determinant , matrices , linear algebra
Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$

1969 IMO Shortlist, 67
Tags: algebra , inequality
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

2021 Princeton University Math Competition, B1
Tags: algebra
Let $x, y$ be distinct positive real numbers satisfying $$\frac{1}{\sqrt{x + y} -\sqrt{x - y}}+\frac{1}{\sqrt{x + y} +\sqrt{x - y}} =\frac{x}{\sqrt{y^3}}.$$ If $\frac{x}{y} =\frac{a+\sqrt{b}}{c}$ for positive integers $a, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.

1987 All Soviet Union Mathematical Olympiad, 453
Tags: square table , sum , table , combinatorics
Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.

1993 Poland - First Round, 1
Tags: system of equations , algebra
Prove that the system of equations $ \begin{cases} \ a^2 - b = c^2 \\ \ b^2 - a = d^2 \\ \end{cases} $ has no integer solutions $a, b, c, d$.

2013 NIMO Problems, 4
Tags: induction
Consider a set of $1001$ points in the plane, no three collinear. Compute the minimum number of segments that must be drawn so that among any four points, we can find a triangle. [i]Proposed by Ahaan S. Rungta / Amir Hossein[/i]

2006 ISI B.Stat Entrance Exam, 2
Tags: quadratic
Suppose that $a$ is an irrational number. (a) If there is a real number $b$ such that both $(a+b)$ and $ab$ are rational numbers, show that $a$ is a quadratic surd. ($a$ is a quadratic surd if it is of the form $r+\sqrt{s}$ or $r-\sqrt{s}$ for some rationals $r$ and $s$, where $s$ is not the square of a rational number). (b) Show that there are two real numbers $b_1$ and $b_2$ such that i) $a+b_1$ is rational but $ab_1$ is irrational. ii) $a+b_2$ is irrational but $ab_2$ is rational. (Hint: Consider the two cases, where $a$ is a quadratic surd and $a$ is not a quadratic surd, separately).

2012 Cuba MO, 6
Tags: geometry
Let $ABC$ be a right triangle at $A$, and let $AD$ be the relative height to the hypotenuse. Let $N$ be the intersection of the bisector of the angle of vertex $C$ with $AD$. Prove that $$AD \cdot BC = AB \cdot DC + BD \cdot AN.$$

1993 AMC 8, 14
Tags:
The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \[\begin{tabular}{|c|c|c|} \hline 1 & & \\ \hline & 2 & A \\ \hline & & B \\ \hline \end{tabular}\] $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

2018 Grand Duchy of Lithuania, 3
Tags: geometry , right angle , circumcenter
The altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at point $H$. Let $F$ be the intersection of the line $AB$ and the line that is parallel to the side BC and goes through the circumcenter of $ABC$. Let $M$ be the midpoint of the segment $AH$. Prove that $\angle CMF = 90^o$

1979 IMO Longlists, 14
Tags: combinatorics
Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds: [b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty. [b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.

Dumbest FE I ever created, 4.
Tags: algebra , function
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$ for all real number $x$ and $y$

2002 Germany Team Selection Test, 3
Tags: decimal representation , number theory , factorial , digit
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2015 South East Mathematical Olympiad, 8
Tags: number theory
For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?

2014 India Regional Mathematical Olympiad, 4
Tags: least common multiple , vieta , symmetry , number theory , system of equations , inequalities , algebra
Find all positive reals $x,y,z $ such that \[2x-2y+\dfrac1z = \dfrac1{2014},\hspace{0.5em} 2y-2z +\dfrac1x = \dfrac1{2014},\hspace{0.5em}\text{and}\hspace{0.5em} 2z-2x+ \dfrac1y = \dfrac1{2014}.\]