Olympiad problems

2016 Iran Team Selection Test, 3

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.

2024 Iranian Geometry Olympiad, 5

Tags: geometry

Cyclic quadrilateral $ABCD$ with circumcircle $\omega$ is given. Let $E$ be a fixed point on segment $AC$. $M$ is an arbitrary point on $\omega$, lines $AM$ and $BD$ meet at a point $P$. $EP$ meets $AB$ and $AD$ at points $R$ and $Q$, respectively, $S$ is the intersection of $BQ,DR$ and lines $MS$ and $AC$ meet at a point $T$. Prove that as $M$ varies the circumcircle of triangle $\bigtriangleup CMT$ passes through a fixed point other than $C$. [i]Proposed by Chunlai Jin - China[/i]

1998 Portugal MO, 6

Tags: algebra , inequalities , sequence , recurrence relation

Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.

2004 All-Russian Olympiad Regional Round, 9.4

Tags: coprime , number theory

Three natural numbers are such that the product of any two of them is divided by the sum of these two numbers. Prove that these three numbers have a common divisor greater than one.

2009 Today's Calculation Of Integral, 480

Tags: integration , inequalities , calculus , calculus computations , geometry

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

LMT Team Rounds 2021+, A17

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Given that the value of \[\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}\] can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Aidan Duncan[/i]

1995 Tuymaada Olympiad, 7

Tags: algebra , continuous function , functional equation

Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$

2006 All-Russian Olympiad, 5

Tags: number theory

Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.

2005 District Olympiad, 4

Tags: irrational number , polynomial , real analysis , algebra , function

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function. a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$. b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.

2004 India IMO Training Camp, 3

Tags: function , combinatorics , geometry

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

2019 CCA Math Bonanza, I10

Tags: absolute value

2005 National Olympiad First Round, 21

Tags: circumcircle , geometry

What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and midpoint of its side $[BC]$? $ \textbf{(A)}\ \dfrac {\sqrt 3}4 \qquad\textbf{(B)}\ \dfrac {\sqrt 5}4 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \dfrac {\sqrt {10}}4 $

2018 Iran Team Selection Test, 5

Tags: geometry

Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$. [i]Proposed by Ali Zamani, Hooman Fattahi[/i]

2014 Argentina Cono Sur TST, 3

Tags: geometry

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and $10$ triangles. Find the maximum number of triangles with the same area that may exist in the division.

2024 Putnam, A5

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Consider the circle $\Omega$ with radius $9$ and center at the origin $(0,\,0)$, and a disk $\Delta$ with radius $1$ and center at $(r,\,0)$, where $0\leq r\leq 8$. Two points $P$ and $Q$ are chosen independently and uniformly at random on $\Omega$. Which value(s) of $r$ minimize the probability that the chord $\overline{PQ}$ intersects $\Delta$?

2007 Greece National Olympiad, 4

Tags: combinatorics

Given a $2007\times 2007$ array of numbers $1$ and $-1$, let $A_{i}$ denote the product of the entries in the $i$th row, and $B_{j}$ denote the product of the entries in the $j$th column. Show that \[A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.\]

1995 Canada National Olympiad, 2

Tags: logarithm , inequalities

Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.

2022 Turkey MO (2nd round), 3

Tags: inequalities

Let $a_1, a_2, \cdots, a_{2022}$ be nonnegative real numbers such that $a_1+a_2+\cdots +a_{2022}=1$. Find the maximum number of ordered pairs $(i, j)$, $1\leq i,j\leq 2022$, satisfying $$a_i^2+a_j\ge \frac 1{2021}.$$

1997 May Olympiad, 2

Tags: geometry

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$

Indonesia MO Shortlist - geometry, g2

Tags: geometry , concyclic , common tangent

It is known that two circles have centers at $P$ and $Q$. Prove that the intersection points of the two internal common tangents of the two circles with their two external common tangents lie on the same circle.

1967 Putnam, B6

Tags: function , derivative , calculus , partial derivatives

Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that $$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$

2012 Hanoi Open Mathematics Competitions, 12

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[b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.

2014 NIMO Problems, 7

Tags: probability , quadratic , ratio

Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$. [i]Proposed by Lewis Chen[/i]

1983 National High School Mathematics League, 7

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$P$ 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? $\text{(A)}9\qquad\text{(B)}17\qquad\text{(C)}1\qquad\text{(D)}5$

2016 Korea Summer Program Practice Test, 7

Tags: geometry , algebra , parallelogram

A infinite sequence $\{ a_n \}_{n \ge 0}$ of real numbers satisfy $a_n \ge n^2$. Suppose that for each $i, j \ge 0$ there exist $k, l$ with $(i,j) \neq (k,l)$, $l - k = j - i$, and $a_l - a_k = a_j - a_i$. Prove that $a_n \ge (n + 2016)^2$ for some $n$.