Olympiad problems

1996 All-Russian Olympiad, 2

Tags: geometry

The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides. [i]D. Tereshin[/i]

2014 National Olympiad First Round, 10

Tags: modular arithmetic

How many non-negative integer triples $(m,n,k)$ are there such that $m^3-n^3=9^k+123$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None of the preceding} $

2016 JBMO Shortlist, 3

Tags: geometry , trapezoid , incenter

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

STEMS 2021 Math Cat B, Q3

Tags: geometry

Let $ABC$ be a triangle with $I$ as incenter.The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle.Make a tangent at $D'$ to incircle.Let it meet $(ABC)$ at $X,Y$ respectively.Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$.Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$

2004 Harvard-MIT Mathematics Tournament, 1

Tags: calculus , trigonometry , limit

Let $f(x)=\sin(\sin(x))$. Evaluate \[ \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} \] at $x=\pi$.

2018 India IMO Training Camp, 3

Tags: combinatorics , convex polygon

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

2008 Indonesia TST, 2

Tags: combinatorics , subset

Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$

2010 Contests, 2

Tags: circumcircle , cyclic quadrilateral , geometry

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

2024 IFYM, Sozopol, 1

Tags: number theory

Given a prime number $ p \geq 3 $ and a positive integer $ m $, find the smallest positive integer $ n $ with the following property: for every positive integer $ a $, which is not divisible by $ p $, the sum of the natural divisors of $ a^n $ greater than 1 is divisible by $ p^m $.

2023 MMATHS, 10

Tags:

Find the number of ordered pairs of integers $(m,n)$ with $0 \le m,n \le 22$ such that $k^2+mk+n$ is not a multiple of $23$ for all integers $k.$

2019 IberoAmerican, 3

Tags: geometry , circumcircle

Let $\Gamma$ be the circumcircle of triangle $ABC$. The line parallel to $AC$ passing through $B$ meets $\Gamma$ at $D$ ($D\neq B$), and the line parallel to $AB$ passing through $C$ intersects $\Gamma$ to $E$ ($E\neq C$). Lines $AB$ and $CD$ meet at $P$, and lines $AC$ and $BE$ meet at $Q$. Let $M$ be the midpoint of $DE$. Line $AM$ meets $\Gamma$ at $Y$ ($Y\neq A$) and line $PQ$ at $J$. Line $PQ$ intersects the circumcircle of triangle $BCJ$ at $Z$ ($Z\neq J$). If lines $BQ$ and $CP$ meet each other at $X$, show that $X$ lies on the line $YZ$.

2023 ELMO Shortlist, A3

Tags: algebra

Does there exist an infinite sequence of integers $a_0$, $a_1$, $a_2$, $\ldots$ such that $a_0\ne0$ and, for any integer $n\ge0$, the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has $n$ distinct real roots? [i]Proposed by Amol Rama and Espen Slettnes[/i]

1976 Bundeswettbewerb Mathematik, 3

Tags: tree , rational , algebra

A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.

2010 Saint Petersburg Mathematical Olympiad, 3

Tags: combinatorics

There are $2009$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $75$ different cities?

2011 Armenian Republican Olympiads, Problem 6

Tags: combinatorics , chess knight , chessboard

Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.

2013 Canadian Mathematical Olympiad Qualification Repechage, 4

Tags: counting , derangement , probability , combinatorics

Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that [i]both[/i] of these events occur: [list] [*] (i) Each person receives exactly one gift; [*] (ii) No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$, then $B$ did not give her gift to $A$).[/list]

2003 Gheorghe Vranceanu, 1

Tags: function , algebra , bijectivity , cardinal

Let $ M $ be a set of nonzero real numbers and $ f:M\longrightarrow M $ be a function having the property that the identity function is $ f+f^{-1} . $ [b]1)[/b] Prove that $ m\in M\iff -m\in M. $ [b]2)[/b] Show that $ f $ is odd. [b]3)[/b] Determine the cardinal of $ M. $

2018 CMIMC CS, 4

Tags: computer science

Consider the grid of numbers shown below. 20 01 96 56 16 37 48 38 64 60 96 97 42 20 98 35 64 96 40 71 50 58 90 16 89 Among all paths that start on the top row, move only left, right, and down, and end on the bottom row, what is the minimum sum of their entries?

2005 Purple Comet Problems, 9

Tags:

Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number

2000 Harvard-MIT Mathematics Tournament, 5

Tags: geometry , 3d geometry

Find all $3$-digit numbers which are the sums of the cubes of their digits.

OMMC POTM, 2024 3

Tags: geometry

Define acute triangle $ABC$ with $AB = AC$ and circumcenter $O$. Define point $D$ inside $ABC$ on the circumcircle of $BOC$. Prove that the distance from $A$ to line $DO$ is half $BD+DC$..

2024 LMT Fall, 25

Tags: speed

Let $a_n$ be a sequence such that $a_1=1$, $a_2=1$, and $a_{n+2}=\tfrac{a_{n+1}a_n}{a_{n+1}+a_n}$. Find the value of \[\sum_{n=1}^\infty \frac{1}{a_n3^n}.\]

1983 IMO Longlists, 63

Tags: trigonometric equation , trigonometry , number theory , equation , trigonometric identities

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

2006 MOP Homework, 1

Tags: circumcircle , geometry

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

1965 Dutch Mathematical Olympiad, 3

Tags: geometry , isosceles , combinatorial geometry

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$, then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$”. Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$, but none with weight $0$.