Olympiad problems

2012 Mathcenter Contest + Longlist, 2 sl9

Tags: algebra , inequalities

Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of . $$a+b+c+\frac{3}{ab+bc+ca}$$ [i](PP-nine)[/i]

1950 Moscow Mathematical Olympiad, 178

Tags: trigonometry , angle

Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles. Is there an angle $x$ such that $$\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C \cdot \cos A} ?$$

2013 ELMO Shortlist, 1

Tags: incenter , circumcircle , inradius , geometry

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

1993 Poland - First Round, 6

Tags: function

The function $f: R \longrightarrow R$ is continuous. Prove that if for every real number $x$, there exists a positive integer $n$, such that $\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1$, then $f(1) = 1$.

2000 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , perpendicular

On side $AB$ of triangle $ABC$, point $D$ is selected. Circle circumscribed around triangle $BCD$, intersects side $AC$ at point $M$, and the circumcircle of triangle $ACD$ intersects the side $BC$ at point $ N$ ($M,N \ne C$). Let $O$ be the circumcenter of the triangle $CMN$. Prove that line $OD$ is perpendicular to side $AB$.

2016 ASDAN Math Tournament, 12

Tags: team test

Let $$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$ Evaluate $$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$

2022 Thailand TSTST, 3

Tags: combinatorics

An odd positive integer $n$ is called pretty if there exists at least one permutation $a_1, a_2,..., a_n$, of $1,2,...,n$, such that all $n$ sums $a_1-a_2+a_3-...+a_n$, $a_2-a_3+a_4-...+a_1$,..., $a_n-a_1+a_2-...+a_{n-1}$ are positive. Find all pretty integers.

1975 Bundeswettbewerb Mathematik, 4

Tags: combinatorics

Two brothers inherited $n$ gold pieces of the total weight $2n$. The weights of the pieces are integers, and the heaviest piece is not heavier than all the other pieces together. Show that if $n$ is even, the brother can divide the inheritance into two parts of equal weight.

2012 Bulgaria National Olympiad, 2

Tags: quadratic , function , algebra

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.

2014 JBMO Shortlist, 4

Tags: number theory , prime number

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2015 JBMO Shortlist, A4

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

2019 USMCA, 22

Tags:

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

Tags:

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

Tags: function , integral , riemann sum , real analysis

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 $\frac{m(m+6)}{4}\leq C(m) \leq \frac{m(m+3)}{2}$.