Olympiad problems

2014 Canada National Olympiad, 2

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

2005 USAMO, 4

Tags: 3d geometry , prism , parallelogram , geometry

Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and still have a stable table? (The table is [i]stable[/i] if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

2016 Dutch IMO TST, 1

Tags: algebra , inequalities

Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$

1994 Chile National Olympiad, 4

Tags: sphere , combinatorial geometry , combinatorics , circles , 3d geometry , geometry

Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

2002 Chile National Olympiad, 1

Tags: number theory

A Metro ticket, which has six digits, is considered a "lucky number" if its six digits are different and their first three digits add up to the same as the last three (A number such as $026134$ is "lucky number"). Show that the sum of all the "lucky numbers" is divisible by $2002$.

2024 AMC 12/AHSME, 9

Tags: analytic geometry , number theory

A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? $ \textbf{(A) }39 \qquad \textbf{(B) }71 \qquad \textbf{(C) }73 \qquad \textbf{(D) }75 \qquad \textbf{(E) }135 \qquad $

2017 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2 + PQ^2$ [color = red]The official problem statement does not have the final period.[/color]

2000 Mongolian Mathematical Olympiad, Problem 2

Tags: geometry , circles

Circles $\omega_1,\omega_2,\omega_3$ with centers $O_1,O_2,O_3$, respectively, are externally tangent to each other. The circle $\omega_1$ touches $\omega_2$ at $P_1$ and $\omega_3$ at $P_2$. For any point $A$ on $\omega_1$, $A_1$ denotes the point symmetric to $A$ with respect to $O_1$. Show that the intersection points of $AP_2$ with $\omega_3$, $A_1P_3$ with $\omega_2$, and $AP_3$ with $A_1P_2$ lie on a line.

2013 USAMTS Problems, 3

Tags: geometry , induction , ratio , inequalities , limit , arithmetic sequence

An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?

2007 Baltic Way, 16

Tags: modular arithmetic , relatively prime , number theory

Let $a$ and $b$ be rational numbers such that $s=a+b=a^2+b^2$. Prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.

1991 Arnold's Trivium, 83

Tags:

Find solutions of the equation $u_t=u_{xxx}+uu_x$ in the form of a traveling wave $u=\varphi(x-ct)$, $\varphi(\pm\infty)=0$.

2016 Bulgaria National Olympiad, Problem 5

Tags: geometry , circumcircle

Let $\triangle {ABC} $ be isosceles triangle with $AC=BC$ . The point $D$ lies on the extension of $AC$ beyond $C$ and is that $AC>CD$. The angular bisector of $ \angle BCD $ intersects $BD$ at point $N$ and let $M$ be the midpoint of $BD$. The tangent at $M$ to the circumcircle of triangle $AMD$ intersects the side $BC$ at point $P$. Prove that points $A,P,M$ and $N$ lie on a circle.

2013 Ukraine Team Selection Test, 8

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

1999 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

2022 BMT, 9

Tags: geometry

Seven spheres are situated in space such that no three centers are collinear, no four centers are coplanar, and every pair of spheres intersect each other at more than one point. For every pair of spheres, the plane on which the intersection of the two spheres lies in is drawn. What is the least possible number of sets of four planes that intersect in at least one point?

2005 Korea National Olympiad, 5

Tags: geometry

Let $P$ be a point that lies outside of circle $O$. A line passes through $P$ and meets the circle at $A$ and $B$, and another line passes through $P$ and meets the circle at $C$ and $D$. The point $A$ is between $P$ and $B$, $C$ is between $P$ and $D$. Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$. Show that $M$ is the midpoint of the segment $DE$, where $M$ is the intersection of lines $PL$ and $DE$.

1984 IMO Longlists, 14

Tags: linear recurrence , algebra , sequence , recurrence relation

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

2010 IMO Shortlist, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

1997 India Regional Mathematical Olympiad, 2

Tags: number theory , greatest common divisor

For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.

2023 ISL, C5

Tags: combinatorics

Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: [list=disc] [*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests. [/list] Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.

2019 Bundeswettbewerb Mathematik, 2

Tags: min , algebra , inequalities

Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$

Estonia Open Senior - geometry, 2005.2.4

Tags: trigonometry , geometric inequality , inequalities , 3d geometry , angle , geometry

Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.

2024 Myanmar IMO Training, 5

Tags: combinatorics , graph theory , easy combinatorics

A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be [i]happy[/i] if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?

1969 IMO Longlists, 25

Tags: number theory , divisibility , frobenius , additive number theory

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

2007 India Regional Mathematical Olympiad, 6

Tags: inequalities

Prove that: [b](a)[/b] $ 5<\sqrt {5}\plus{}\sqrt [3]{5}\plus{}\sqrt [4]{5}$ [b](b)[/b] $ 8>\sqrt {8}\plus{}\sqrt [3]{8}\plus{}\sqrt [4]{8}$ [b](c)[/b] $ n>\sqrt {n}\plus{}\sqrt [3]{n}\plus{}\sqrt [4]{n}$ for all integers $ n\geq 9 .$ [b][Weightage 16/100][/b]