Olympiad problems

1992 IMO Shortlist, 15

Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

1996 Czech and Slovak Match, 3

Tags: inequalities , solid geometry , pyramid , geometry , 3d geometry

The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a$\sqrt{17}$. Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.

2009 National Olympiad First Round, 31

Tags: inequalities

For all $ |x| \ge n$, the inequality $ |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2$ holds. Integer $ n$ can be at least ? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5$

2013 Tournament of Towns, 4

Tags: coloring , sum , product , algebra , combinatorics

On a circle, there are $1000$ nonzero real numbers painted black and white in turn. Each black number is equal to the sum of two white numbers adjacent to it, and each white number is equal to the product of two black numbers adjacent to it. What are the possible values of the total sum of $1000$ numbers?

2015 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorics , graph theory

In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly $100$ roads. We call $10$ roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.

2006 Chile National Olympiad, 4

Tags: number theory , perfect square , perfect cube

Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .

2017 Kosovo Team Selection Test, 5

Tags: geometry

Given triangle $ABC$. Let $P$, $Q$, $R$, be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$, $BC$, $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$. Prove that $AP'$, $BQ'$, and $CR'$ are concurrent.

2021 Peru Cono Sur TST., P2

Tags: number theory

For each positive integer $k$ we denote by $S(k)$ the sum of its digits, for example $S(132)=6$ and $S(1000)=1$. A positive integer $n$ is said to be $\textbf{fascinating}$ if it holds that $n = \frac{k}{S(k)}$ for some positive integer $k$. For example, the number $11$ is $\textbf{fascinating}$ since $11 = \frac{198}{S(198)} ($since $\frac{198}{S(198)}=\frac{198}{1+9+8}=\frac{198}{18} = 11)$. Prove that there exists a positive integer less than $2021$ and that it is not $\textbf{fascinating}$.

2010 Argentina Team Selection Test, 4

Tags: geometry , rhombus , combinatorics

Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$. $A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started. If $A$ starts the game, determine which player has a winning strategy.

2010 239 Open Mathematical Olympiad, 3

Tags: combinatorics

Grisha wrote $n$ different natural numbers, the sum of which does not exceed $S$. The saboteur added to each of them a number from the half-interval $[0, 1)$. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than $1$. At what minimum $S$ can Grisha ensure that the sabotage will definitely not be succeeded?

Kvant 2022, M2710

Tags: combinatorics , board , geometry

We are given an $(n^2-1)\times(n^2-1)$ checkered board. A set of $n{}$ cells is called [i]progressive[/i] if the centers of the cells lie on a straight line and form $n-1$ equal intervals. Find the number of progressive sets. [i]Proposed by P. Kozhevnikov[/i]

2011 Indonesia TST, 2

Tags: combinatorics , geometry , probability , acute

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

2010 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , noncollinear

In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points

2010 Today's Calculation Of Integral, 560

Tags: calculus , integration , function , geometry , geometric transformation , rotation , logarithm

Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$. (1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis. (2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis. Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.

2024 Sharygin Geometry Olympiad, 10.4

Tags: geo , geometry

Let $I$ be the incenter of a triangle $ABC$. The lines passing through $A$ and parallel to $BI, CI$ meet the perpendicular bisector to $AI$ at points $S, T$ respectively. Let $Y$ be the common point of $BT$ and $CS$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $YA^*$ lies on the excircle of the triangle touching the side $BC$.

2018 Moscow Mathematical Olympiad, 4

Tags: geometry , circumcircle

$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.

2014 JBMO TST - Macedonia, 1

Tags: integer , inequality

Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$

2023 Romania National Olympiad, 4

Tags: continuous function , real analysis , differentiable function

We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that \[ g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. \] a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$ b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$

2016 AIME Problems, 9

Tags: sequence

The sequences of positive integers $1,a_2,a_3,\ldots$ and $1,b_2,b_3,\ldots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.

2018 MIG, 8

Tags:

The set of natural numbers are arranged as so: $$\begin{array}{ccccccccc} & & & & 1 & & & &\\ & & & 2 & 3 & 4 & & &\\ & & 5 & 6 & 7 & 8 & 9 &\\ & 10 & 11 & 12 & 13 & 14 & 15 & 16 &\\ 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25\\ & & & & \vdots & & & & \end{array}$$ so that each row has $2$ more numbers in it, and the rows are centered. What is the number under $49$? $\textbf{(A) }60\qquad\textbf{(B) }61\qquad\textbf{(C) }62\qquad\textbf{(D) }63\qquad\textbf{(E) }64$

2010 Purple Comet Problems, 14

Tags: algebra , polynomial

There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$

2019 Purple Comet Problems, 11

Tags: number theory

Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square.

2016 Saudi Arabia BMO TST, 3

Tags: number theory

Let $d$ be a positive integer. Show that for every integer $S$, there exist a positive integer $n$ and a sequence $a_1, ..., a_n \in \{-1, 1\}$ such that $S = a_1(1 + d)^2 + a_2(1 + 2d)^2 + ... + a_n(1 + nd)^2$.

2013 Tournament of Towns, 7

Tags: game strategy , game

On a table, there are $11$ piles of ten stones each. Pete and Basil play the following game. In turns they take $1, 2$ or $3$ stones at a time: Pete takes stones from any single pile while Basil takes stones from different piles but no more than one from each. Pete moves first. The player who cannot move, loses. Which of the players, Pete or Basil, has a winning strategy?

1993 Baltic Way, 17

Tags: vector , geometry

Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?