Olympiad problems

2012 India PRMO, 12

Tags: algebra

If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$, where $m$ and $n$ are positive integers, what is the value of $m+n$?

2011 Bosnia Herzegovina Team Selection Test, 1

Tags: geometry , incenter , circumcircle , geometric transformation , homothety , ratio , analytic geometry

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

2018 BAMO, 4

Tags: number theory , algebra , perfect cube , integer , geometry , 3d geometry

(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

2005 Today's Calculation Of Integral, 65

Tags: calculus , integration , calculus computations

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

1997 Taiwan National Olympiad, 4

Tags: number theory

Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.

2010 IMO Shortlist, 5

Tags: function , algebra , functional equation

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

2019 Romania National Olympiad, 2

Tags: geometry , trapezoid , complex number geometry , discrete geometry , combinatorics

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

LMT Theme Rounds, 12

Tags:

A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed. [i]Proposed by Nathan Ramesh

2021 Balkan MO Shortlist, C5

Tags:

Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves: (a) He clears every piece of rubbish from a single pile. (b) He clears one piece of rubbish from each pile. However, every evening, a demon sneaks into the warehouse and performs exactly one of the following moves: (a) He adds one piece of rubbish to each non-empty pile. (b) He creates a new pile with one piece of rubbish. What is the first morning when Angel can guarantee to have cleared all the rubbish from the warehouse?

JOM 2015 Shortlist, N4

Tags: number theory , diophantine equation

Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$

2011 China Western Mathematical Olympiad, 4

Tags: geometric transformation , homothety , angle bisector , geometry

In a circle $\Gamma_{1}$, centered at $O$, $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$. A circle $\Gamma_{2}$, centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$, and also tangent to $AB$ at $G$ and $CD$ at $H$. A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$. The line $EF$ intersects $l$ at $M$. Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$

2014 Belarusian National Olympiad, 3

Tags: geometry , perimeter , inscribed , geometric inequality

The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

2019 Iran MO (3rd Round), 2

Tags: geometry , circumcircle

In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.

1969 IMO Longlists, 32

Tags: 3d geometry , sphere , dissection , combinatorics , geometry

$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.

2015 Hanoi Open Mathematics Competitions, 13

Tags: number theory , relatively prime , exponential

Let $m$ be given odd number, and let $a, b$ denote the roots of equation $x^2 + mx - 1 = 0$ and $c = a^{2014} + b^{2014}$ , $d =a^{2015} + b^{2015}$ . Prove that $c$ and $d$ are relatively prime numbers.

2021 Princeton University Math Competition, A7

Tags: algebra

Consider the following expression $$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$ Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).

1958 AMC 12/AHSME, 12

Tags: logarithm

If $ P \equal{} \frac{s}{(1 \plus{} k)^n}$ then $ n$ equals: $ \textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad \textbf{(B)}\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad \textbf{(C)}\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \\ \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}$

2005 Polish MO Finals, 1

Tags: inequalities , function , number theory

Find all triplets $(x,y,n)$ of positive integers which satisfy: \[ (x-y)^n=xy \]

2008 Bulgaria Team Selection Test, 1

Tags: number theory

For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?

2005 ISI B.Math Entrance Exam, 5

Tags: function , inequalities , analytic geometry , graphing lines , slope , calculus

Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

2013 CHMMC (Fall), 4

Tags: algebra

Let $$A =\frac12 +\frac13 +\frac15 +\frac19,$$ $$B =\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{2 \cdot 9}+\frac{1}{3 \cdot 5}+\frac{1}{3 \cdot 9} +\frac{1}{5 \cdot 9},$$ $$C =\frac{1}{2 \cdot 3 \cdot 5} + \frac{1}{2 \cdot 3 \cdot 9} + \frac{1}{2 \cdot 5 \cdot 9} +\frac{1}{3 \cdot 5 \cdot 9}.$$ Compute the value of $A + B + C$.

2023 BMT, 20

Tags: number theory

Call a positive integer, $n$, [i]ready [/i] if all positive integer divisors of $n$ have a ones digit of either $1$ or $3$. Let S be the sum of all positive integer divisors of $32!$ that are ready. Compute the remainder when S is divided by $131$.

Novosibirsk Oral Geo Oly VII, 2019.4

Tags: square , geometry , isosceles , collinear

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2024 Belarus Team Selection Test, 3.4

Tags: number theory , combinatorics

Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$ [i]N. Sheshko, D. Zmiaikou[/i]

2020 Romanian Master of Mathematics Shortlist, N1

Tags: number theory , function

Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct. (For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.) [i]Poland, Wojciech Nadara[/i]