Olympiad problems

2018 PUMaC Team Round, 5

Tags:

There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have $$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$ Find $a+b+c+d$.

1991 China National Olympiad, 5

Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)

2011 Junior Balkan MO, 1

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

2015 Portugal MO, 6

Tags: combinatorics , combinatorial geometry , geometry

For what values of $n$ is it possible to mark $n$ points on the plane so that each point has at least three other points at distance $1$?

2007 Mongolian Mathematical Olympiad, Problem 4

Tags: algebra , inequalities

Let $ a,b,c>0$. Prove that $ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$

2013 Saint Petersburg Mathematical Olympiad, 1

Tags: trigonometry , floor function , calculus , integration , algebra

Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $. F. Petrov

2014 BMT Spring, 5

Tags: geometry , number theory

Call two regular polygons supplementary if the sum of an internal angle from each polygon adds up to $180^o$. For instance, two squares are supplementary because the sum of the internal angles is $90^o + 90^o = 180^o$. Find the other pair of supplementary polygons. Write your answer in the form $(m, n)$ where m and n are the number of sides of the polygons and $m < n$.

2000 Harvard-MIT Mathematics Tournament, 8

Tags: geometry

A sphere is inscribed inside a pyramid with a square as a base whose height is $\frac{\sqrt{15}}{2}$ times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?

2014 All-Russian Olympiad, 1

Tags: trigonometry , algebra

Does there exist positive $a\in\mathbb{R}$, such that \[|\cos x|+|\cos ax| >\sin x +\sin ax \] for all $x\in\mathbb{R}$? [i]N. Agakhanov[/i]

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10

Tags: modular arithmetic

The number of pairs of integers $ (m,n)$ satisfying the equation \[ m^3 \plus{} 6m^2 \plus{} 5m \equal{} 27n^3 \plus{} 9n^2 \plus{} 9n \plus{} 1\] is $ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{Infinitely many}$

Estonia Open Junior - geometry, 1995.1.2

Tags: geometry , rectangle , circles

Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.

2017 Junior Balkan Team Selection Tests - Moldova, Problem 8

Tags: combinatorics , grid , operation , optimization , swap , ordering

The bottom line of a $2\times 13$ rectangle is filled with $13$ tokens marked with the numbers $1, 2, ..., 13$ and located in that order. An operation is a move of a token from its cell into a free adjacent cell (two cells are called adjacent if they have a common side). What is the minimum number of operations needed to rearrange the chips in reverse order in the bottom line of the rectangle?

2022 Rioplatense Mathematical Olympiad, 2

Tags: combinatorics

Eight teams play a rugby tournament in which each team plays exactly one match against each of the remaining seven teams. In each match, if it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $2$ points and the loser gets $0$ points. After the tournament it was observed that each of the eight teams had a different number of points and that the number of points of the winner of the tournament was equal to the sum of the number of points of the last four teams. Give an example of a tournament that satisfies this conditions, indicating the number of points obtained by each team and the result of each match.

2008 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , geometric transformation , combinatorics

A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other. Fresh translation; slightly reworded.

2021 Portugal MO, 4

Tags: combinatorics

Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,

2021 Kyiv City MO Round 1, 10.1

Tags: inequalities

Prove the following inequality: $$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$ [i]Proposed by Oleksii Masalitin[/i]

2016 Turkey EGMO TST, 1

Tags: inequalities

Prove that \[ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) \] for all positive real numbers $x, y, z$.

2009 Dutch IMO TST, 1

Tags: induction , number theory , easy

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$. Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a [i]stump[/i] of $n$. Let $T(n)$ be the sum of all stumps of $n$. Prove that $n=S(n)+9T(n)$.

2003 Greece National Olympiad, 3

Tags: ratio , geometry

Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$ Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)

2017 Regional Olympiad of Mexico Northeast, 2

Tags: geometry , angle bisector

Let $ABC$ be a triangle and let $N$ and $M$ be the midpoints of $AB$ and $CA$, respectively. Let $H$ be the foot of altitude from $A$. The circumcircle of $ABH$ intersects $MN$ at $P$, with $P$ and $M$ on the same side relative to $N$, and the circumcircle of $ACH$ intersects $MN$ at $Q$, with $Q$ and $N$ on the same side relative to $M$. $BP$ and $CQ$ intersect at $X$. Prove that $AX$ is the angle bisector of $\angle CAB$.

2016 AIME Problems, 14

Tags:

Centered at each lattice point in the coordinate plane are a circle of radius $\tfrac{1}{10}$ and a square with sides of length $\tfrac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0, 0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of the circles. Find $m + n$.

1988 Spain Mathematical Olympiad, 5

Tags: partition , combinatorics , square

A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture. [img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img] Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)

2007 Putnam, 1

Tags: algebra , polynomial , function

Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$

2015 BMT Spring, 10

Tags: combinatorics

A partition of a positive integer $n$ is a summing $n_1+\ldots+n_k=n$, where $n_1\ge n_2\ge\ldots\ge n_k$. Call a partition [i]perfect[/i] if every $m\le n$ can be represented uniquely as a sum of some subset of the $n_i$'s. How many perfect partitions are there of $n=307$?

2008 Finnish National High School Mathematics Competition, 1

Tags: number theory

Foxes, wolves and bears arranged a big rabbit hunt. There were $45$ hunters catching $2008$ rabbits. Every fox caught $59$ rabbits, every wolf $41$ rabbits and every bear $40$ rabbits. How many foxes, wolves and bears were there in the hunting company?