Olympiad problems

2024 Durer Math Competition Finals, 1

Tags: fraction , algebra

Describe all ordered sets of four real numbers $(a, b, c, d)$ for which the values $a + b, b + c, c + d, d + a$ are all non-zero and \[\frac{a+2b+3c}{c+d}=\frac{b+2c+3d}{d+a}=\frac{c+2d+3a}{a+b}=\frac{d+2a+3b}{b+c}.\]

2023 Bulgarian Autumn Math Competition, 8.3

Tags: number theory

Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$ for all positive integers $n$.

2004 Junior Tuymaada Olympiad, 3

Tags: perpendicular , arc , collinear , geometry

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

1956 Moscow Mathematical Olympiad, 324

Tags: point , combinatorial geometry , combinatorics , distance , geometry , circles

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

1996 AMC 12/AHSME, 26

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An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely: (a) the selection of four red marbles; (b) the selection of one white and three red marbles; (c) the selection of one white, one blue, and two red marbles; and (d) the selection of one marble of each color. What is the smallest number of marbles satisfying the given condition? $\text{(A)}\ 19 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 46 \qquad \text{(D)}\ 69\qquad \text{(E)}\ \text{more than 69}$

2017 Canadian Mathematical Olympiad Qualification, 7

Tags: combinatorics , number theory

Given a set $S_n = \{1, 2, 3, \ldots, n\}$, we define a [i]preference list[/i] to be an ordered subset of $S_n$. Let $P_n$ be the number of preference lists of $S_n$. Show that for positive integers $n > m$, $P_n - P_m$ is divisible by $n - m$. [i]Note: the empty set and $S_n$ are subsets of $S_n$.[/i]

JOM 2015 Shortlist, A2

Tags: inequality , inequalities

Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.

1972 Bundeswettbewerb Mathematik, 3

Tags: induction , combinatorics

$2^{n-1}$ subsets are choosen from a set with $n$ elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.

2018 USAMTS Problems, 4:

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Find, with proof, all ordered pairs of positive integers $(a, b)$ with the following property: there exist positive integers $r$, $s$, and $t$ such that for all $n$ for which both sides are defined, [center]${{n\choose{a}}\choose{b}}=r {{n+s}\choose{t}}$ .[/center]

1998 IMO Shortlist, 6

Tags: combinatorial geometry , coloring , point set , combinatorics

Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of $k$ colors, in such a way that for any $k$ of the ten points, there are $k$ segments each joining two of them and no two being painted with the same color. Determine all integers $k$, $1\leq k\leq 10$, for which this is possible.

1969 Dutch Mathematical Olympiad, 1

Tags: remainder , number theory

Determine the smallest $n$ such that $n \equiv (a - 1)$ mod $a$ for all $a \in \{2,3,..., 10\}$.

2023 Kurschak Competition, 2

Tags: analytic geometry , grid , graph theory , combinatorics

Let $n\geq 2$ be a positive integer. We call a [i]vertex[/i] every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an [i]edge[/i], if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is [i]vital[/i] for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.

2007 Estonia Team Selection Test, 4

Tags: angle , angle chasing , square , geometry

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

2011 Sharygin Geometry Olympiad, 2

Tags: midpoint , tangential quadrilateral , cyclic quadrilateral , geometry

Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.

2016 All-Russian Olympiad, 7

Tags: incircle , geometry , circumcircle

In triangle $ABC$,$AB<AC$ and $\omega$ is incirle.The $A$-excircle is tangent to $BC$ at $A^\prime$.Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$.The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$.Prove that the sum $XY+XZ$ not depends to point $X$.(Mitrofanov)

2009 JBMO TST - Macedonia, 4

Tags: combinatorics , algorithm , rectangle , induction , geometry

In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?

1991 Mexico National Olympiad, 2

Tags: number theory , palindrome

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.

2012 Today's Calculation Of Integral, 822

Tags: integration , calculus computations , calculus , limit

For $n=0,\ 1,\ 2,\ \cdots$, let $a_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}(x-n)\}\ dx,$ $b_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}\}\ dx.$ Find $\lim_{n\to\infty} \sum_{k=0}^n (a_k-b_k).$

2017 Turkey Team Selection Test, 4

Tags: graph theory , combinatorics

Each two of $n$ students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of $n$.

2011 All-Russian Olympiad, 4

Tags: invariant , graph theory , combinatorics

A $2010\times 2010$ board is divided into corner-shaped figures of three cells. Prove that it is possible to mark one cell in each figure such that each row and each column will have the same number of marked cells. [i]I. Bogdanov & O. Podlipsky[/i]

1981 Romania Team Selection Tests, 1.

Tags: polynomial , algebra

Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$. [i]Mircea Becheanu[/i]

2002 Paraguay Mathematical Olympiad, 1

Tags: combinatorics

There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist .

2002 Canada National Olympiad, 3

Tags: am-gm , inequalities , algebra

Prove that for all positive real numbers $a$, $b$, and $c$, \[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \] and determine when equality occurs.

2020 China Team Selection Test, 6

Tags: graph theory , combinatorics

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

2023 HMNT, 15

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Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by $20$ and increases the larger number by $23,$ only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.