Olympiad problems

2024 Ukraine National Mathematical Olympiad, Problem 3

Tags: geometry , orthocenter

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. [i]Proposed by Anton Trygub[/i]

1986 IMO Shortlist, 11

Tags: geometry , point set , combinatorial geometry , straight lines , IMO Shortlist

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

2003 China National Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Find all integer triples $(a,m,n)$ such that $a^m+1|a^n+203$ where $a,m>1$. [i]Chen Yonggao[/i]

2014 AMC 8, 11

Tags: Asymptote , analytic geometry , probability , AMC

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$

2015 IFYM, Sozopol, 6

Tags: algebra , functional equation

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ : $f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.

1990 Tournament Of Towns, (258) 2

Tags: combinatorics , weighings

We call a collection of weights (each weighing an integer value) basic if their total weight equals $500$ and each object of integer weight not greater than $500$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equal value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $500$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

2012-2013 SDML (Middle School), 5

Tags: abstract algebra

What is the hundreds digit of the sum below? $$1+12+123+1234+12345+123456+1234567+12345678+123456789$$ $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2000 All-Russian Olympiad, 1

Tags: quadratics , algebra unsolved , algebra

Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$.

Novosibirsk Oral Geo Oly VIII, 2019.3

Tags: ratio , paper folding , folding , geometry , square

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

1995 Tournament Of Towns, (450) 6

Tags: geometry , combinatorial geometry , combinatorics

Can it happen that $6$ parallelepipeds, no two of which have common points, are placed in space so that there is a point outside of them from which no vertex of a parallelepiped is visible? (The parallelepipeds are not transparent.) (V Proizvolov)

1990 IMO Longlists, 4

Tags: function , trigonometry , inequalities unsolved , inequalities

Find the minimal value of the function \[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]

Champions Tournament Seniors - geometry, 2008.4

Tags: geometry , 3D geometry , sphere , pyramid , Champions Tournament

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.

2000 Tuymaada Olympiad, 4

Tags: inequalities , logarithms , inequalities proposed

Prove for real $x_1$, $x_2$, ....., $x_n$, $0 < x_k \leq {1\over 2}$, the inequality \[ \left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right). \]

1953 Polish MO Finals, 2

Tags: geometry , Locus , rectangle , inscribed

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

2016 Postal Coaching, 4

Tags: polynomial , algebra

Let $f$ be a polynomial with real coefficients and suppose $f$ has no nonnegative real root. Prove that there exists a polynomial $h$ with real coefficients such that the coefficients of $fh$ are nonnegative.

2011 Tournament of Towns, 1

Tags: combinatorial geometry , geometry , convex polyhedron , polyhedron , similar triangles

The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

2023 Moldova Team Selection Test, 6

Tags: algebra

Show that if $2023$ real numbers $x_1,x_2,\dots,x_{2023}$ satisfy $x_1\geq x_2\geq\dots\geq x_{2023}\geq0,$ then $$x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.$$ When does the equality take place?

2014 Cuba MO, 5

Tags: combinatorics , number theory

The number 2013 is written on a blackboard. Two players participate, alternating in turns, in the next game. A movement consists in changing the number that is on the board for the difference of this number and one of its divisors. The player who writes a zero loses. Which of the two players can guarantee victory?

2013 AIME Problems, 3

Tags: Gauss , AMC , algebra , AIME

A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.

2022 Brazil EGMO TST, 5

Tags: Sequence , recurrence relation , algebra

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

1968 AMC 12/AHSME, 26

Tags: AMC

Let $S=2+4+6+ \cdots +2N$, where $N$ is the smallest positive integer such that $S>1,000,000$. Then the sum of the digits of $N$ is: $\textbf{(A)}\ 27 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

2017 Kosovo Team Selection Test, 1

Tags: number theory

Find all positive integers $(a, b)$, such that $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.

2010 National Olympiad First Round, 7

Tags: geometry , similar triangles

A frog is at the center of a circular shaped island with radius $r$. The frog jumps $1/2$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that yields the frog can never fall into the water? $ \textbf{(A)}\ \frac{\sqrt 5}{3} \qquad\textbf{(B)}\ \frac{\sqrt {13}}{5} \qquad\textbf{(C)}\ \frac{\sqrt {19}}{6} \qquad\textbf{(D)}\ \frac{1}{\sqrt 2} \qquad\textbf{(E)}\ \frac34 $

II Soros Olympiad 1995 - 96 (Russia), 10.6

Tags: geometry , inradius , equal circles

On sides $BC$, $CA$ and $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are taken, respectively, so that the radii of the circles inscribed in triangles $A_1BC_1$, $AB_1C_1$ and $A_1B_1C$ are equal to each other and equal to $r$. The radius of the circle inscribed in triangle $A_1B_1C_1$ is equal to $r_1$. Find the radius of the circle inscribed in triangle $ABC$.

2005 CHKMO, 4

Tags: function , number theory , prime numbers , combinatorics unsolved , combinatorics

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.