Olympiad problems

2024 Sharygin Geometry Olympiad, 15

Tags: inradius , ratio , circumcircle , geometry , algebra

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

2016 Balkan MO Shortlist, N4

Tags: number theory , polynomial

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

2017 Online Math Open Problems, 6

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A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

2024 Euler Olympiad, Round 2, 1

Tags: euler , diophantine equation , number theory

Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]

1999 AIME Problems, 1

Tags: number theory , arithmetic sequence

Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.

1971 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine

Prove that $(0,1)$, $(0, -1)$,$( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$

2008 Sharygin Geometry Olympiad, 2

Tags: geometry

(F.Nilov) Given quadrilateral $ ABCD$. Find the locus of points such that their projections to the lines $ AB$, $ BC$, $ CD$, $ DA$ form a quadrilateral with perpendicular diagonals.

2015 NIMO Problems, 1

Tags: geometry

Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$ while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$. [i]Proposed by David Altizio[/i]

Russian TST 2022, P2

Tags: combinatorics , construction , number theory

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

2011 Balkan MO Shortlist, A2

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Given an integer $n \geq 3$, determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition \begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}

2023 Stars of Mathematics, 1

Tags: prime number , perfect square , number theory

Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

2017 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry

Given a scalene triangle $ABC$ with $\angle B=130^{\circ}$. Let $H$ be the foot of altitude from $B$. $D$ and $E$ are points on the sides $AB$ and $BC$, respectively, such that $DH=EH$ and $ADEC$ is a cyclic quadrilateral. Find $\angle{DHE}$.

1980 Putnam, A1

Tags: function , conic , parabola , algebra , polynomial

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

1997 National High School Mathematics League, 1

Tags: geometry

Two circles with different radius $O_1$ and $O_2$ are both tangent to a larger circle $O$, tangent points are $S,T$. Note that intersections of $O_1$ and $O_2$ are $M,N$, prove that the sufficient and necessary condition of $OM\perp MN$ is $S,N,T$ are colinear.

2018 BMT Spring, 3

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$2018$ people (call them $A, B, C, \ldots$) stand in a line with each permutation equally likely. Given that $A$ stands before $B$, what is the probability that $C$ stands after $B$?

2009 Today's Calculation Of Integral, 515

Tags: derivative , calculus computations , integration , calculus , trigonometry

Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$. Note that you are not allowed to solve in using partial differentiation here.

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: geometry , angle

Angle $A$ in triangle $ABC$ is equal to $a$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at a point $M$ different from $A$. Find the angle $\angle BMC$.

2006 Federal Math Competition of S&M, Problem 4

Tags: game , combinatorics

There are $n$ coins aligned in a row. In each step, it is allowed to choose a coin with the tail up (but not one of the outermost markers), remove it and reverse the closest coin to the left and the closest coin to the right of it. Initially, all the coins have tails up. Prove that one can achieve the state with only two coins remaining if and only if $n-1$ is not divisible by $3$.

2020 Iran Team Selection Test, 6

Tags: number theory , prime number

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

2007 Croatia Team Selection Test, 1

Tags: calculus , integration , number theory

Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

2023 Azerbaijan IZhO TST, 1

Tags: geometry

In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

2010 China Team Selection Test, 3

Tags: number theory

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

2011 Tournament of Towns, 2

Tags: right triangle , geometry

On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.

2018 CMIMC Number Theory, 5

Tags: binomial coefficient , number theory

It is given that there exist unique integers $m_1,\ldots, m_{100}$ such that \[0\leq m_1 < m_2 < \cdots < m_{100}\quad\text{and}\quad 2018 = \binom{m_1}1 + \binom{m_2}2 + \cdots + \binom{m_{100}}{100}.\] Find $m_1 + m_2 + \cdots + m_{100}$.

2009 Moldova National Olympiad, 8.3

Tags: geometry , circles , parallel

The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.