Olympiad problems

2018 ASDAN Math Tournament, 5

Tags: geometry

In pentagon $ABCDE$, $BE$ intersects $AC$ and $AD$ at $F$ and $G$, respectively. Suppose that $A[\vartriangle AF G] = A[\vartriangle BCF] = A[\vartriangle DEG] = 16$, where$ A[\vartriangle AF G]$ denotes the area of $\vartriangle AF G$. Next, suppose that $BF = 4$, $F G = 5$, and $GE = 6$. Compute $A[ABCDE]$.

1999 Junior Balkan Team Selection Tests - Moldova, 4

Tags: geometry , area of a triangle , equilateral , ratio , area

Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.

1985 Iran MO (2nd round), 1

Tags: trigonometry , induction , strong induction , algebra

Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.

2018 Iran Team Selection Test, 3

Tags: geometry , pole and polar , pappus

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

2011 All-Russian Olympiad, 3

Tags: graph theory , combinatorics

The graph $G$ is not $3$-coloured. Prove that $G$ can be divided into two graphs $M$ and $N$ such that $M$ is not $2$-coloured and $N$ is not $1$-coloured. [i]V. Dolnikov[/i]

1998 Brazil Team Selection Test, Problem 3

Tags: function , polynomial , algebra

Find all functions $f: \mathbb N \to \mathbb N$ for which \[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\] holds for all positive integers $n$.

Kvant 2019, M2557

Tags: combinatorics , graph theory , probability , cycle

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

1999 All-Russian Olympiad Regional Round, 9.1

Tags: number theory , digit

All natural numbers from $1$ to $N$, $ N \ge 2$ are written out in a certain order in a circle. Moreover, for any pair of neighboring numbers there is at least one digit appearing in the decimal notation of each of them. Find the smallest possible value of $N$.

2008 Cuba MO, 3

Tags: number theory , integer

Prove that there are infinitely many ordered pairs of positive integers $(m, n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer.

2014 Saudi Arabia Pre-TST, 2.1

Tags: divide , divisible , number theory

Prove that $2014$ divides $53n^{55}- 57n^{53} + 4n$ for all integer $n$.

2022 LMT Fall, 7

Tags: combinatorics

A regular hexagon is split into $6$ congruent equilateral triangles by drawing in the $3$ main diagonals. Each triangle is colored $1$ of $4$ distinct colors. Rotations and reflections of the figure are considered nondistinct. Find the number of possible distinct colorings.

2008 Harvard-MIT Mathematics Tournament, 2

Tags: symmetry

Let $ S \equal{} \{1,2,\ldots,2008\}$. For any nonempty subset $ A\in S$, define $ m(A)$ to be the median of $ A$ (when $ A$ has an even number of elements, $ m(A)$ is the average of the middle two elements). Determine the average of $ m(A)$, when $ A$ is taken over all nonempty subsets of $ S$.

2001 AIME Problems, 6

Tags: probability , analytic geometry , counting , distinguishability , number theory

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1981 Romania Team Selection Tests, 1.

Tags: algebra

Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$. Show that \[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\] [i]Ioan Tomescu[/i]

2012 Kazakhstan National Olympiad, 3

Tags: geometry , rectangle , combinatorics

The cell of a $(2m +1) \times (2n +1)$ board are painted in two colors - white and black. The unit cell of a row (column) is called [i]dominant[/i] on the row (the column) if more than half of the cells that row (column) have the same color as this cell. Prove that at least $m + n-1$ cells on the board are dominant in both their row and column.

2005 AMC 12/AHSME, 4

Tags:

At the beginning of the school year, Lisa’s goal was to earn an A on at least $ 80\%$ of her $ 50$ quizzes for the year. She earned an A on $ 22$ of the first $ 30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2010 Sharygin Geometry Olympiad, 1

Tags: geometry , angle bisector , angle chasing , altitude , angle

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

1999 Estonia National Olympiad, 4

Tags: odd , combinatorics , chessboard

Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.

2022 IOQM India, 10

Tags: algebra , polynomial

Suppose that $P$ is the polynomial of least degree with integer coefficients such that $$P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})$$Find $P(2)$.

2007 Iran Team Selection Test, 3

Tags: circumcircle , geometry

$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.

1964 All Russian Mathematical Olympiad, 042

Tags: number theory , power

Prove that for no natural $m$ a number $m(m+1)$ is a power of an integer.

2006 Victor Vâlcovici, 3

Tags: group theory

Consider the operation $ * $ on $ \mathbb{R} $ defined as $ x*y=x\sqrt{1+y^2}+y\sqrt{1+x^2} . $ Prove that the real numbers form a group under this operation and it's isomorphic with the additive group of real numbers.

2023 Kazakhstan National Olympiad, 1

Tags: geometry

The $C$-excircle of a triangle $ABC$ touches $AB, AC, BC$ at $M, N, K$. The points $P, Q$ lie on $NK$ so that $AN=AP, BK=BQ$. Prove that the circumradius of $\triangle MPQ$ is equal to the inradius of $\triangle ABC$.

2011 Brazil Team Selection Test, 4

Tags: modular arithmetic , divisibility , number theory

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

1987 IMO Shortlist, 12

Tags: circumcircle , incenter , collinearity , geometry

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]