Olympiad problems

2019 USMCA, 22

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Find the largest real number $\lambda$ such that \[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\] for all real numbers $a_1, \ldots, a_{2019}$. The coefficients on the right-hand side are $1$ for all terms except $a_{1009}a_{1010}$ and $a_{1010}a_{1011}$, which have coefficient $\lambda$.

2020 Harvard-MIT Mathematics Tournament, 1

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Let $DIAL$, $FOR$, and $FRIEND$ be regular polygons in the plane. If $ID=1$, find the product of all possible areas of $OLA$. [i]Proposed by Andrew Gu.[/i]

2011 District Olympiad, 3

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function. [b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing. [b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.

1993 India Regional Mathematical Olympiad, 3

Tags: congruent triangles , geometry

Suppose $A_1, A_2, A_3, \ldots, A_{20}$is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon?

1986 IMO Longlists, 41

Tags: circumcircle , trigonometry , symmetry , geometry

Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$

1962 All-Soviet Union Olympiad, 12

Tags: number theory , algebra , divisibility

Given unequal integers $x, y, z$ prove that $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y- z)(z-x)$.

2010 Brazil National Olympiad, 2

Tags: polynomial , ceiling function , number theory , prime number , algebra

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2016 Costa Rica - Final Round, N2

Tags: prime , number theory

Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

2018 Peru Iberoamerican Team Selection Test, P8

Tags: combinatorics

A new chess piece named $ mammoth $ moves like a bishop l (that is, in a diagonal), but only in 3 of the 4 possible directions. Different $ mammoths $ in the board may have different missing addresses. Find the maximum number of $ mammoths $ that can be placed on an $ 8 \times 8 $ chessboard so that no $ mammoth $ can be attacked by another.

LMT Team Rounds 2021+, 11

Tags: combinatorics

The LHS Math Team is going to have a Secret Santa event! Nine members are going to participate, and each person must give exactly one gift to a specific recipient so that each person receives exactly one gift. But to make it less boring, no pairs of people can just swap gifts. The number of ways to assign who gives gifts to who in the Secret Santa Exchange with these constraints is $N$. Find the remainder when $N$ is divided by $1000$.

2004 Estonia National Olympiad, 1

Tags: gcd , lcm , number theory

Find all triples of positive integers $(x, y, z)$ satisfying $x < y < z$, $gcd(x, y) = 6, gcd(y, z) = 10, gcd(z, x) = 8$ and $lcm(x, y,z) = 2400$.

1998 Belarusian National Olympiad, 7

Tags: point , combinatorial geometry , combinatorics

On the plane $n+1$ points are marked, no three of which lie on one straight line. For what natural $k$ can they be connected by segments so that for any $n$ marked points there are exactly $k$ segments with ends at these points?

2023 Bundeswettbewerb Mathematik, 3

Tags: area of a triangle , geometry , angle

Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$. If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?

2007 Romania National Olympiad, 4

Tags: combinatorics , inequalities

a) For a finite set of natural numbers $S$, denote by $S+S$ the set of numbers $z=x+y$, where $x,y\in S$. Let $m=|S|$. Show that $|S+S|\leq \frac{m(m+1)}{2}$. b) Let $m$ be a fixed positive integer. Denote by $C(m)$ the greatest integer $k\geq 1$ for which there exists a set $S$ of $m$ integers, such that $\{1,2,\ldots,k\}\subseteq S\cup(S+S)$. For example, $C(3)=8$, with $S=\{1,3,4\}$. Show that $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$.

2020 IMO, 1

Tags: geometry

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: \[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\] Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$. [i]Proposed by Dominik Burek, Poland[/i]

2006 AMC 10, 18

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Let $ a_1, a_2, ...$ be a sequence for which \[a_1 \equal{} 2\,\hspace{.2in}a_2 \equal{} 3\, \hspace{.2in}\text{and}\hspace{.2in}a_n \equal{} \frac {a_{n \minus{} 1}}{a_{n \minus{} 2}} \text{ for each positive integer } n \ge 3.\]What is $ a_{2006}$? $\textbf{(A) } \frac 12 \qquad \textbf{(B) } \frac 23 \qquad \textbf{(C) } \frac 32 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } 3$

2015 CCA Math Bonanza, I2

Tags: factorial

The operation $*$ is defined by the following: $a*b=a!-ab-b.$ Compute the value of $5*8.$ [i]2015 CCA Math Bonanza Individual Round #2[/i]

2012 Online Math Open Problems, 13

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A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$. [i]Author: Ray Li[/i]

2023 Brazil Cono Sur TST, 1

Tags: number theory

Let $n = p_1p_2 \dots p_k$ be the product of distinct primes $p_1, p_2, \dots , p_k$, with $k > 1$. Find all $n$ such that $n$ is multiple of $p_1 - 1, p_2 - 1, \dots , p_k - 1$.

1962 All Russian Mathematical Olympiad, 018

Tags: geometry , geometric construction , triangle construction

Given two sides of the triangle. Construct that triangle, if medians to those sides are orthogonal.

1999 French Mathematical Olympiad, Problem 1

Tags: geometry , 3d geometry

What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.

2018 Cono Sur Olympiad, 1

Tags: geometry

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

2016 Grand Duchy of Lithuania, 2

Tags: combinatorics

During a school year $44$ competitions were held. Exactly $7$ students won in each of the competitions. For any two competitions, there exists exactly $1$ student who won in both competitions. Is it true that there exists a student who won all of the competitions?

2010 Kyrgyzstan National Olympiad, 4

Tags: geometry

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

2020 Moldova EGMO TST, 4

Tags: geometry , incenter , angle bisector

The incircle of triangle $ABC$ touches $AC$ and $BC$ respectively $P$ and $Q$. Let $N$ and $M$ be the midpoints of the sides $AC$ and $BC$ respectively.$AM$ and $BP$,$BN$ and $AQ$ intersects at the points $X$ and $Y$ respectively. If the points $C,X$ and $Y$ are collinear , then prove that $CX$ is the angle bisector of $\angle ACB$.