Olympiad problems

2011 APMO, 2

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

1988 Federal Competition For Advanced Students, P2, 5

Tags: geometric transformation , reflection , geometry , circumcircle , incenter

The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.

2011 Romania Team Selection Test, 3

Tags: probabilistic method , combinatorics , geometry

Given a set $L$ of lines in general position in the plane (no two lines in $L$ are parallel, and no three lines are concurrent) and another line $\ell$, show that the total number of edges of all faces in the corresponding arrangement, intersected by $\ell$, is at most $6|L|$. [i]Chazelle et al., Edelsbrunner et al.[/i]

1995 IMC, 1

Tags: matrix , linear algebra

Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$. Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$. Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$´s for eigenvalues.

2002 Balkan MO, 3

Tags: parallelogram , reflection , geometric transformation , rectangle , homothety , circumcircle , geometry

Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.

2020 Korea - Final Round, P4

Tags: number theory

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? \[ 2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots \] If so, determine all such pairs; if not, prove that it is impossible.

2004 USAMTS Problems, 3

Tags:

A set is $reciprocally\ whole$ if its elements are distinct integers greater than 1 and the sum of the reciprocals of all these elements is exactly 1. Find a set $S$, as small as possible, that contains two reciprocally whole subsets, $I$ and $J$, which are distinct, but not necessarily disjoint (meaning they may share elements, but they may not be the same subset). Prove that no set with fewer elements than $S$ can contain two reciprocally whole subsets.

2011 National Olympiad First Round, 35

Tags: calculus , derivative , function , inequalities

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

2014 Harvard-MIT Mathematics Tournament, 23

Tags: expected value , absolute value , probability

Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.

2022 Bulgarian Spring Math Competition, Problem 10.3

Tags: permutations with restrictions , combinatorics

A permutation $\sigma$ of the numbers $1,2,\ldots , 10$ is called $\textit{bad}$ if there exist integers $i, j, k$ which satisfy \[1 \leq i < j < k \leq 10 \quad \text{ and }\quad \sigma(j) < \sigma(k) < \sigma(i)\] and $\textit{good}$ otherwise. Find the number of $\textit{good}$ permutations.

III Soros Olympiad 1996 - 97 (Russia), 11.6

Tags: combinatorial geometry , combinatorics , geometry

What is the largest number of obtuse triangles that can be composed of $16$ different segments (each triangle is composed of three segments), if the largest of these segments does not exceed twice the smallest?

2005 Canada National Olympiad, 1

Tags: combinatorics

An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$. Determine the value of $ f(2005)$.

1989 Irish Math Olympiad, 3

Tags: algebra , function

A function $f$ is defined on the natural numbers $\mathbb{N}$ and satisfies the following rules: (a) $f(1)=1$; (b) $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$ for all $n\in \mathbb{N}$. Calculate the maximum value $m$ of the set $\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}$, and determine the number of natural numbers $n$, with $1\le n\le 1989$, that satisfy the equation $f(n)=m$.

2023 Junior Balkan Team Selection Tests - Moldova, 5

Tags: geometry , number theory

The positive integers $ a, b, c $ are the lengths of the sides of a right triangle. Prove that $abc$ is divisible by $60$.

MOAA Gunga Bowls, 2021.2

Tags:

Add one pair of brackets to the expression \[1+2\times 3+4\times 5+6\] so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make? [i]Proposed by Nathan Xiong[/i]

2011 Iran Team Selection Test, 10

Tags: calculus , inequalities

Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality \[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\] holds.

2021 Sharygin Geometry Olympiad, 10-11.2

Tags: midpoint , angle bisector , geometry , concurrency

Let $ABC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.

Geometry Mathley 2011-12, 8.4

Tags: concurrent circles , tangent circle , circles , orthocenter

Let $ABC$ a triangle inscribed in a circle $(O)$ with orthocenter $H$. Two lines $d_1$ and $d_2$ are mutually perpendicular at $H$. Let $d_1$ meet $BC,CA,AB$ at $X_1, Y_1,Z_1$ respectively. Let $A_1B_1C_1$ be a triangle formed by the line through $X_1$ perpendicular to $BC$, the line through $Y_1$ perpendicular to CA, the line through $Z_1$ perpendicular perpendicular to $AB$. Triangle $A_2B_2C_2$ is defined in the same manner. Prove that the circumcircles of triangles $A_1B_1C_1$ and $A_2B_2C_2$ touch each other at a point on $(O)$. Nguyễn Văn Linh

2023 India Regional Mathematical Olympiad, 3

Tags: number theory

For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \text { and }(s(m))^2=n . $$

2022 All-Russian Olympiad, 6

Tags: combinatorics , geometry

Given is natural number $n$. Sasha claims that for any $n$ rays in space, no two of which have a common point, he will be able to mark on these rays $k$ points lying on one sphere. What is the largest $k$ for which his statement is true?

2013 Poland - Second Round, 1

Tags: trinomial , divisibility , number theory

Let $b$, $c$ be integers and $f(x) = x^2 + bx + c$ be a trinomial. Prove, that if for integers $k_1$, $k_2$ and $k_3$ values of $f(k_1)$, $f(k_2)$ and $f(k_3)$ are divisible by integer $n \neq 0$, then product $(k_1 - k_2)(k_2 - k_3)(k_3 - k_1)$ is divisible by $n$ too.

2014 Sharygin Geometry Olympiad, 4

Tags: concyclic , geometry

Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be? (A. Garkavyj, A. Sokolov )

2010 Singapore Junior Math Olympiad, 3

Tags: positive integer , sum , combinatorics

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

Russian TST 2017, P2

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

1974 IMO Longlists, 37

Tags: trigonometry

Let $a, b$, and $c$ denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side $a$ and then propelled toward side $b$ with direction defined by the angle $\theta$. For what values of $\theta$ will the ball strike the sides $b, c, a$ in that order?