Olympiad problems

2023 Polish Junior Math Olympiad First Round, 4.

Tags: combinatorics

Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that: – every red number is a sum of two distinct blue numbers; and – every blue number is a difference between two red numbers. Determine the smallest possible value of $n$ for which such a coloring exists.

2018 India IMO Training Camp, 1

Tags: sorting , distance , extremal combinatorics , radius , string , combinatorics

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

1988 IMO Longlists, 50

Tags: rectangle , combinatorics , geometry

Prove that the numbers $A,B$ and $C$ are equal, where: - $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles. - $B=$ number of sequences of ones and twos that add up to $n$ - $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and - $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$

2023 Euler Olympiad, Round 1, 4

Tags: euler , algebra

Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. [i]Proposed by Gogi Khimshiashvili, Georgia[/i]

2001 Polish MO Finals, 2

Tags: geometry , parallelogram , geometric transformation , reflection , ratio , analytic geometry , angle bisector

Let $ABCD$ be a parallelogram and let $K$ and $L$ be points on the segments $BC$ and $CD$, respectively, such that $BK\cdot AD=DL\cdot AB$. Let the lines $DK$ and $BL$ intersect at $P$. Show that $\measuredangle DAP=\measuredangle BAC$.

2018 Bulgaria EGMO TST, 3

Tags: induction , function , algebra

Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds: \[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

2009 F = Ma, 12

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Batman, who has a mass of $\text{M = 100 kg}$, climbs to the roof of a $\text{30 m}$ building and then lowers one end of a massless rope to his sidekick Robin. Batman then pulls Robin, who has a mass of $\text{m = 75 kg}$, up the roof of the building. Approximately how much total work has Batman done after Robin is on the roof? (A) $\text{60 J}$ (B) $\text{7} \times \text{10}^3 \text{J}$ (C) $\text{5} \times \text{10}^4 \text{J}$ (D) $\text{600 J}$ (E) $\text{3} \times \text{10}^4 \text{J}$

1983 Swedish Mathematical Competition, 6

Tags: system of equations , system , algebra

Show that the only real solution to \[\left\{ \begin{array}{l} x(x+y)^2 = 9 \\ x(y^3 - x^3) = 7 \\ \end{array} \right. \] is $x = 1$, $y = 2$.

1998 Putnam, 5

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Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

2005 Germany Team Selection Test, 1

Tags: number theory , equation , divisor

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2017 Kazakhstan NMO, Problem 4

Tags: geometry

The acute triangle $ABC$ $(AC> BC)$ is inscribed in a circle with the center at the point $O$, and $CD$ is the diameter of this circle. The point $K$ is on the continuation of the ray $DA$ beyond the point $A$. And the point $L$ is on the segment $BD$ $(DL> LB)$ so that $\angle OKD = \angle BAC$, $\angle OLD = \angle ABC$. Prove that the line $KL$ passes through the midpoint of the segment $AB$.

Kvant 2025, M2828

Tags: algebra , polynomial

Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps? [i]M. Didin[/i]

2004 Bosnia and Herzegovina Team Selection Test, 1

Tags: collinear , touching , geometry , circles

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

1969 IMO Longlists, 30

Tags: number theory , sophie germain identity , composite number

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

2014 Canada National Olympiad, 2

Tags: floor function , combinatorics

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

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