Olympiad problems

2020 Regional Olympiad of Mexico Center Zone, 6

Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$

1989 French Mathematical Olympiad, Problem 3

Tags: geometry , 3D geometry , tetrahedron , inequalities , geometric inequality , Geometric Inequalities

Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$, the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$.

1998 National Olympiad First Round, 26

Tags:

How many ordered integer pairs $ \left(x,y\right)$ are there satisfying following equation: \[ y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.\] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 1998 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ \text{Infinitely many}$

1963 Putnam, A6

Tags: Putnam , conics , ellipse , geometry

Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$, and let $AB$ and $CD$ be any two other chords through $M$. If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$, prove that $M$ is the midpoint of the segment $PQ.$

PEN D Problems, 8

Tags: Congruences

Characterize the set of positive integers $n$ such that, for all integers $a$, the sequence $a$, $a^2$, $a^3$, $\cdots$ is periodic modulo $n$.

1973 Miklós Schweitzer, 7

Tags: advanced fields , advanced fields unsolved

Let us connect consecutive vertices of a regular heptagon inscribed in a unit circle by connected subsets (of the plane of the circle) of diameter less than $ 1$. Show that every continuum (in the plane of the circle) of diameter greater than $ 4$, containing the center of the circle, intersects one of these connected sets. [i]M. Bognar[/i]

1966 IMO Longlists, 43

Tags: combinatorics , Coloring , graph theory , Ramsey Theory , Extremal combinatorics , IMO Longlist , IMO Shortlist

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color. [b]a.)[/b] Show that: [i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue. [i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.) [b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

2003 Italy TST, 2

Tags: analytic geometry , induction , combinatorics proposed , combinatorics , Hi

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares. $(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard? $(b)$ When it is possible, find the minimum number of trominoes needed.

2009 May Olympiad, 2

Tags: geometry , Angle Chasing , isosceles , Equilateral Triangle , convex quadrilateral

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

MOAA Individual Speed General Rounds, 2021.8

Tags: MOAA 2021 , speed

Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

2019 Thailand TST, 3

Tags: IMO Shortlist , combinatorics , IMO shortlist 2018

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

2015 Bundeswettbewerb Mathematik Germany, 3

Tags: Coloring , combinatorics , counting

Each of the positive integers $1,2,\dots,n$ is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number $x$ and the smallest number larger than $x$ colored in the same color as $x$ always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.

2024 Brazil Cono Sur TST, 2

Tags: geometry

Let $ABC$ be a triangle with $AB < AC < BC$ and $\Gamma$ its circumcircle. Let $\omega_1$ be the circle with center $B$ and radius $AC$ and $\omega_2$ the circle with center $C$ and radius $AB$. The circles $\omega_1$ and $\omega_2$ intersect at point $E$ such that $A$ and $E$ are on opposite sides of the line $BC$. The circles $\Gamma$ and $\omega_1$ intersect at point $F$ and the circles $\Gamma$ and $\omega_2$ intersect at point $G$ such that the points $F$ and $G$ are on the same side as $E$ in relation to the line $BC$. With $K$ being the point such that $AK$ is a diameter of $\Gamma$, prove that $K$ is circumcenter of triangle $EFG$.

2011 Tournament of Towns, 3

Tags: combinatorics

From the $9 \times 9$ chessboard, all $16$ unit squares whose row numbers and column numbers are both even have been removed. Disect the punctured board into rectangular pieces, with as few of them being unit squares as possible.

2015 Brazil National Olympiad, 3

Tags: number theory proposed , number theory , prime factorization

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its [i]false derived[/i] is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.

2013 USA TSTST, 7

Tags: linear algebra , matrix , rotation , geometry , geometric transformation , vector , calculus

A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

2012 AMC 8, 21

Tags: geometry , 3D geometry , AMC

Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2 \qquad \textbf{(B)}\hspace{.05in}10 \qquad \textbf{(C)}\hspace{.05in}10\sqrt2 \qquad \textbf{(D)}\hspace{.05in}50 \qquad \textbf{(E)}\hspace{.05in}50\sqrt2 $

1984 IMO Longlists, 30

Tags: geometric sequence , number theory unsolved , number theory

Decide whether it is possible to color the $1984$ natural numbers $1, 2, 3, \cdots, 1984$ using $15$ colors so that no geometric sequence of length $3$ of the same color exists.

1979 AMC 12/AHSME, 10

Tags: geometry , Asymptote , trapezoid , AMC

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is $\textbf{(A) }6\qquad\textbf{(B) }2\sqrt{6}\qquad\textbf{(C) }\frac{8\sqrt{3}}{3}\qquad\textbf{(D) }3\sqrt{3}\qquad\textbf{(E) }4\sqrt{3}$

2022 Novosibirsk Oral Olympiad in Geometry, 1

Tags: geometry , square

Cut a square with three straight lines into three triangles and four quadrilaterals.

2013 Today's Calculation Of Integral, 877

Tags: calculus , integration , limit , trigonometry , calculus computations , Definite integral , limits

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

Kvant 2022, M2684

Tags: algebra

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

1996 German National Olympiad, 5

Tags: construction , geometric construction , geometry

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.

2020 LIMIT Category 1, 13

Tags: geometry , limit

On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.

2006 MOP Homework, 3

Tags: number theory , greatest common divisor , inequalities , number theory unsolved

Prove that the following inequality holds with the exception of finitely many positive integers $n$: $\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.