Olympiad problems

2014 Paenza, 6

(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then: \[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\] (b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then: \[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]

2023 BAMO, 5

Tags: combinatorics

A [i]lattice point[/i] in the plane is a point with integer coordinates. Let $T$ be a triangle in the plane whose vertice are lattice points, but with no other lattice points on its sides. Furthermore, suppose $T$ contains exactly four lattice points in its interior. Prove that these four points lie on a straight line.

2019 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , incenter

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

2024 239 Open Mathematical Olympiad, 4

Tags: combinatorics

Let $n$ be a positive integer greater than $1$ and let us call an arbitrary set of cells in a $n\times n$ square $\textit{good}$ if they are the intersection cells of several rows and several columns, such that none of those cells lie on the main diagonal. What is the minimum number of pairwise disjoint $\textit{good}$ sets required to cover the entire table without the main diagonal?

2023 Macedonian Mathematical Olympiad, Problem 5

Tags: combinatorics

There are $n$ boys and $n$ girls sitting around a circular table, where $n>3$. In every move, we are allowed to swap the places of $2$ adjacent children. The [b]entropy[/b] of a configuration is the minimal number of moves such that at the end of them each child has at least one neighbor of the same gender. Find the maximal possible entropy over the set of all configurations. [i]Authored by Viktor Simjanoski[/i]

2003 China Western Mathematical Olympiad, 3

Tags: modular arithmetic , pigeonhole principle , number theory

Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$, in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$.

2014 Saint Petersburg Mathematical Olympiad, 2

Tags: combinatorics

There are $40$ points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than $3^{39}$

2014 Flanders Math Olympiad, 4

Tags: algebra , polynomial

Let $P(x)$ be a polynomial of degree $5$ and suppose that a and b are real numbers different from zero. Suppose the remainder when $P(x)$ is divided by $x^3 + ax + b$ equals the remainder when $P(x)$ is divided by $x^3 + ax^2 + b$. Then determine $a + b$.

2007 IMC, 1

Tags: quadratic , algebra , polynomial , modular arithmetic

Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.

1989 IMO Shortlist, 21

Tags: geometry , 3d geometry , tetrahedron , angle , geometric inequality

Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$

1999 Estonia National Olympiad, 3

Tags: geometry , incircle , area

The incircle of the triangle $ABC$, with the center $I$ , touches the sides $AB, AC$ and $BC$ in the points $K, L$ and $M$ respectively. Points $P$ and $Q$ are taken on the sides $AC$ and $BC$ respectively, such that $|AP| = |CL|$ and $|BQ| = |CM|$. Prove that the difference of areas of the figures $APIQB$ and $CPIQ$ is equal to the area of the quadrangle $CLIM$

2010 VJIMC, Problem 1

Tags: summation , convergence

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

2000 German National Olympiad, 4

Tags: system of equations , algebra

Find all nonnegative solutions $(x,y,z)$ to the system $$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\ \sqrt{x+z}+\sqrt{y} = 7 \\ \sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$

2014 AMC 12/AHSME, 1

Tags:

What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$ ${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$

MOAA Team Rounds, 2022.9

Tags: algebra

Emily has two cups $A$ and $B$, each of which can hold $400$ mL, A initially with $200$ mL of water and $B$ initially with $300$ mL of water. During a round, she chooses the cup with more water (randomly picking if they have the same amount), drinks half of the water in the chosen cup, then pours the remaining half into the other cup and refills the chosen cup to back to half full. If Emily goes for $20$ rounds, how much water does she drink, to the nearest integer?

2017 District Olympiad, 3

Tags: inequalities

Let $ a $ be a positive real number. Prove that $$ a^{\sin x}\cdot (a+1)^{\cos x}\ge a,\quad\forall x\in \left[ 0,\frac{\pi }{2} \right] . $$

2018 Indonesia MO, 8

Tags: geometry

Let $I, O$ be the incenter and circumcenter of the triangle $ABC$ respectively. Let the excircle $\omega_A$ of $ABC$ be tangent to the side $BC$ on $N$, and tangent to the extensions of the sides $AB, AC$ on $K, M$ respectively. If the midpoint of $KM$ lies on the circumcircle of $ABC$, prove that $O, I, N$ are collinear.

2007 Chile National Olympiad, 4

Tags: combinatorics

$31$ guests at a party sit in equally spaced chairs around a round table , but they have not noticed that there are cards with the names of the guests on the stalls. (a) Assuming they have been so unlucky that no one is in the room which corresponds to him, show that it is possible to get at least two people to stay in their correct position, without anyone getting up from their seat, turning the table. (b) Show a configuration where exactly one guest is in his assigned place and where in no way that the table is turned it is possible to achieve that at least two remain right.

2017 Online Math Open Problems, 3

Tags:

The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $i/p$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$. How many pages did you turn in? [i]Proposed by Tristan Shin[/i]

2010 Purple Comet Problems, 2

Tags:

Three boxes each contain four bags. Each bag contains five marbles. How many marbles are there altogether in the three boxes?

2013 Romania National Olympiad, 1

Tags: induction , arithmetic sequence , algebra

A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

2020 Korean MO winter camp, #5

Tags: geometry

$\square ABCD$ is a quadrilateral with $\angle A=2\angle C <90^\circ$. $I$ is the incenter of $\triangle BAD$, and the line passing $I$ and perpendicular to $AI$ meets rays $CB$ and $CD$ at $E,F$ respectively. Denote $O$ as the circumcenter of $\triangle CEF$. The line passing $E$ and perpendicular to $OE$ meets ray $OF$ at $Q$, and the line passing $F$ and perpendicular to $OF$ meets ray $OE$ at $P$. Prove that the circle with diameter $PQ$ is tangent to the circumcircle of $\triangle BCD$.

II Soros Olympiad 1995 - 96 (Russia), 9.4

Tags: number theory , algebra

$100$ schoolchildren took part in the Mathematical Olympiad. $4$ tasks were proposed. The first problem was solved by $90$ people, the second by $80$, the third by $70$ and the fourth by $60$. However, no one solved all the problems. Students who solved both the third and fourth questions received an award. How many students were awarded?

2021 Israel National Olympiad, P5

Tags: algebra

Solve the following equation in positive numbers. $$(2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1)=828567056280801$$

2002 Moldova National Olympiad, 4

Tags: geometry , geometric transformation , reflection , symmetry

The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.