Olympiad problems

2021 CMIMC, 2.1

Tags: combinatorics

We have a $9$ by $9$ chessboard with $9$ kings (which can move to any of $8$ adjacent squares) in the bottom row. What is the minimum number of moves, if two pieces cannot occupy the same square at the same time, to move all the kings into an $X$ shape (a $5\times5$ region where there are $5$ kings along each diagonal of the $X$, as shown below)? \begin{tabular}{ c c c c c } O & & & & O \\ & O & & O & \\ & & O & & \\ & O & & O & \\ O & & & & O \\ \end{tabular} [i]Proposed by David Tang[/i]

2019 South East Mathematical Olympiad, 7

Tags: geometry

Let $ABCD$ be a given convex quadrilateral in a plane. Prove that there exist a line with four different points $P,Q,R,S$ on it and a square $A’B’C’D’$ such that $P$ lies on both line $AB$ and $A’B’,$ $Q$ lies on both line $BC$ and $B’C’,$ $R$ lies on both line $CD$ and $C’D’,$ $S$ lies on both line $DA$ and $D’A’.$

2004 Alexandru Myller, 3

Tags: altitude , geometry

Let $ ABC $ be a right triangle in $ A, $ and let be a point $ D $ on $ BC. $ The bisectors of $ \angle ADB $ and $ \angle ADC $ intersect $ AB $ and $ AC $ (respectively) in $ M $ and $ N $ (respectively). Show that the small angle between $ BC $ and $ MN $ is equal to $ \frac{1}{2}\cdot\left| \angle ABC -\angle BCA \right| $ if and only if $ D $ is the feet of the perpendicular from $ A. $ [i]Bogdan Enescu[/i]

2006 India Regional Mathematical Olympiad, 2

Tags: number theory

If $ a$ and $ b$ are natural numbers such that $ a\plus{}13b$ is divisible by $ 11$ and $ a\plus{}11b$ is divisible by $ 13$, then find the least possible value of $ a\plus{}b$.

1995 IMO Shortlist, 5

Tags: combinatorics , graph theory , extremal combinatorics

At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

2010 Contests, 3

Tags: combinatorics , graph theory , vertex degree

At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?

2019 Saudi Arabia Pre-TST + Training Tests, 4.3

Tags: geometry , circumcenter , angle bisector

Let $ABC$ be a triangle, let $D$ be the touch point of the side $BC$ and the incircle of the triangle $ABC$, and let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the bisector of the angle $BAC$.

MOAA Team Rounds, TO4

Tags: algebra , team

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

2000 JBMO ShortLists, 3

Tags: floor function , number theory

Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

2009 Ukraine National Mathematical Olympiad, 4

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds \[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\] [b]Note.[/b] $P^2(k)=\left( P(k) \right)^2.$

2002 Iran MO (3rd Round), 22

Tags: combinatorics

15000 years ago Tilif ministry in Persia decided to define a code for $n\geq2$ cities. Each code is a sequence of $0,1$ such that no code start with another code. We know that from $2^{m}$ calls from foreign countries to Persia $2^{m-a_{i}}$ of them where from the $i$-th city (So $\sum_{i=1}^{n}\frac1{2^{a_{i}}}=1$). Let $l_{i}$ be length of code assigned to $i$-th city. Prove that $\sum_{i=1}^{n}\frac{l_{i}}{2^{i}}$ is minimum iff $\forall i,\ l_{i}=a_{i}$

2012 German National Olympiad, 1

Tags: number theory , sequence , recursive , composite

Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.

Indonesia MO Shortlist - geometry, g8

Tags: concyclic , geometry

Given a circle centered at point $O$, with $AB$ as the diameter. Point $C$ lies on the extension of line $AB$ so that $B$ lies between $A$ and $C$, and the line through $C$ intersects the circle at points $D$ and $E$ (where $D$ lies between $C$ and $E$). $OF$ is the diameter of the circumcircle of triangle $OBD$, and the extension of the line $CF$ intersects the circumcircle of triangle $OBD$ at point $G$. Prove that the points $O, A, E, G$ lie on a circle.

1992 IMO Longlists, 82

Tags: algebra , polynomial , product

Let $f(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m$ and $g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n$ be two polynomials with real coefficients such that for each real number $x, f(x)$ is the square of an integer if and only if so is $g(x)$. Prove that if $n +m > 0$, then there exists a polynomial $h(x)$ with real coefficients such that $f(x) \cdot g(x) = (h(x))^2.$ [hide="Remark."]Remark. The original problem stated $g(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n$, but I think the right form of the problem is what I wrote.[/hide]

1991 All Soviet Union Mathematical Olympiad, 535

Tags: system of equations , diophantine , number theory

Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

2024 Malaysian IMO Training Camp, 4

Tags: algebra

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ [i]Proposed by Wong Jer Ren[/i]

1954 AMC 12/AHSME, 9

Tags:

A point $ P$ is outside a circle and is $ 13$ inches from the center. A secant from $ P$ cuts the circle at $ Q$ and $ R$ so that the external segment of the secant $ PQ$ is $ 9$ inches and $ QR$ is $ 7$ inches. The radius of the circle is: $ \textbf{(A)}\ 3" \qquad \textbf{(B)}\ 4" \qquad \textbf{(C)}\ 5" \qquad \textbf{(D)}\ 6" \qquad \textbf{(E)}\ 7"$

2023 AMC 12/AHSME, 17

Tags: geometry , arithmetic sequence

Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$? $\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$

Novosibirsk Oral Geo Oly IX, 2016.6

Tags: geometry , equal segments , equilateral

An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.

2020 BMT Fall, Tie 2

Tags: geometry

Quadrilateral $ABCD$ is cyclic with $AB = CD = 6$. Given that $AC = BD = 8$ and $AD+3 = BC$, the area of $ABCD$ can be written in the form $\frac{p\sqrt{q}}{r}$, where $p, q$, and $ r$ are positive integers such that $p$ and $ r$ are relatively prime and that $q$ is square-free. Compute $p + q + r$.

Gheorghe Țițeica 2024, P2

Tags: real analysis , derivative

Find all monotonic and twice differentiable functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f''+4f+3f^2+8f^3=0.$$

1995 AMC 12/AHSME, 24

Tags: logarithm

There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that \[A \log_{200} 5 + B \log_{200} 2 = C. \] What is $A+B+C$? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2010 Today's Calculation Of Integral, 595

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

2018 CMIMC CS, 2

Tags: computer science

Consider the natural implementation of computing Fibonacci numbers: \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{FIB}(n)$: \\ 2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\ 3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$ \end{tabular} When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur?

2018 Malaysia National Olympiad, A4

Tags: geometry , octagon , area

Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.