Olympiad problems

2022 VIASM Summer Challenge, Problem 3

Tags: combinatorics

Given an isosceles trapezoid and draw one of its diagonals (so we get $2$ triangles). On each triangle we get, there is an ant walking along the edge. There speed are equal and unchanged. These $2$ ants also didn't change their walking directions and their directions are opposite. Prove that: for all initial locations of the ants, they will meet each other at some time.

1999 Putnam, 2

Tags: Putnam , algebra , polynomial , college contests

Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),f_2(x),\ldots,f_k(x)$ such that \[p(x)=\sum_{j=1}^k(f_j(x))^2.\]

2024 Ukraine National Mathematical Olympiad, Problem 7

Tags: number theory

Prove that there exist infinitely many positive integers that can't be represented in form $a^{bc} - b^{ad}$, where $a, b, c, d$ are positive integers and $a, b>1$. [i]Proposed by Anton Trygub, Oleksii Masalitin[/i]

2009 Today's Calculation Of Integral, 486

Tags: calculus , integration , trigonometry , analytic geometry , geometry , calculus computations

Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$ Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions. (1) Express the coordiante of $ H$ in terms of $ t$. (2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$. (3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: algebra , trigonometry

Solve the equation $$arc \sin (\sin x) + arc \cos (\cos x)=0$$

2023 AMC 12/AHSME, 11

Tags: geometry , trapezoid , AMC , AMC 12 , area , 2023 AMC , 2023 AMC 12B

What is the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other? $ \textbf{(A) }\frac 54 \qquad \textbf{(B) } \frac 87 \qquad \textbf{(C)} \frac{5\sqrt2}4 \qquad \textbf{(D) } \frac 32 \qquad \textbf{(E) } \frac{3\sqrt3}4 $

1982 Bundeswettbewerb Mathematik, 2

Tags: Triangle , Equilateral , projection , space , geometry

Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.

2025 Euler Olympiad, Round 1, 1

Tags: algebra , Euler Olympiad

Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board. [i]Proposed by Giorgi Arabidze, Georgia[/i]

2022 District Olympiad, P1

Tags: romania , monoid , college contests

Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that [list=a] [*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty; [*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$; [*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$ [/list] [i]Mathematical Gazette[/i]

2005 Estonia National Olympiad, 5

Tags: Perfect Square , number theory

Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?

2014 IMO Shortlist, C9

Tags: IMO Shortlist , combinatorics , combinatorial geometry

There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or $\textit{vice versa}$. Suppose that Turbo’s path entirely covers all circles. Prove that $n$ must be odd. [i]Proposed by Tejaswi Navilarekallu, India[/i]

2015 SDMO (High School), 5

Tags:

Let $A$ be a finite set of points in the coordinate plane. Suppose that $A$ has $n\geq3$ points. Given any $a$ in $A$, the horizontal and vertical lines through $a$ define four [i]closed[/i] quadrants centered at $a$. For any real number $\alpha$, call a point $a$ in $A$ $\alpha$-good if there are two diagonally opposite closed quadrants centered at $a$ that each contain at least $\alpha n$ points from $A$. Show that there is some $a$ in $A$ that is $\frac{1}{8}$-good.

2003 District Olympiad, 1

Tags: 3D geometry , geometry , angles , planes , Equilateral

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

2008 Moldova National Olympiad, 9.1

Tags: quadratics , parameterization , algebra , quadratic formula , algebra unsolved

Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)\equal{}(m^2\plus{}m\plus{}1)x^2\minus{}2(m^2\plus{}1)x\plus{}m^2\minus{}m\plus{}1,$ where $ m \in \mathbb R$. 1) Find the fixed common point of all this parabolas. 2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.

2010 Tournament Of Towns, 1

Tags:

The exchange rate in a Funny-Money machine is $s$ McLoonies for a Loonie or $\frac{1}{s}$ Loonies for a McLoonie, where $s$ is a positive real number. The number of coins returned is rounded off to the nearest integer. If it is exactly in between two integers, then it is rounded up to the greater integer. $(a)$ Is it possible to achieve a one-time gain by changing some Loonies into McLoonies and changing all the McLoonies back to Loonies? $(b)$ Assuming that the answer to $(a)$ is "yes", is it possible to achieve multiple gains by repeating this procedure, changing all the coins in hand and back again each time?

2020 Thailand TSTST, 2

Tags: Sequence , Arithmetic Functions , number theory

For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$, $$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$ and, for each positive integer $k$ such that $b_{k-1}\geq 2$, $$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$ It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$. Prove that the numbers of terms in the two sequences are always equal.

2019 Caucasus Mathematical Olympiad, 8

Tags: number theory

Determine if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_{101}$, $b_1$, $b_2$, \ldots, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1,2,\ldots,101\}$ the sum $\sum\limits_{i\in S}a_i$ divides $\left( 100!+\sum\limits_{i\in S}b_i \right)$.

1961 All Russian Mathematical Olympiad, 007

Tags: combinatorics , table

Given some $m\times n$ table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

2007 Denmark MO - Mohr Contest, 3

Tags: combinatorics

A cunning dragon guards a princess. To overcome the dragon and to win the princess you must solve the following task: The dragon has in some of the fields $i$ the columned hall (see figure) with the numbers $1-8$. Even in the rest of the fields you can place numbers $9-36$. The numbers $1-36$ must be arranged so that any turn that starts with one enters from either the south or the west, and ends up going out towards either the north or east, goes through at least one number from the $5$ table. (On the figure are north, south, east and west indicated by N, S, E and W). Georg wants to win the princess. Is it possible to be done? [img]https://cdn.artofproblemsolving.com/attachments/0/7/2ad1b7f944847ee6d3c614ea6c2656865808e7.png[/img]

1986 Brazil National Olympiad, 3

Tags: geometry , combinatorial geometry , angles

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

1894 Eotvos Mathematical Competition, 3

Tags: geometry , arithmetic sequence

The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.

2005 Romania National Olympiad, 3

Tags: linear algebra , matrix , calculus , function , number theory , greatest common divisor , least common multiple

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.

1988 Federal Competition For Advanced Students, P2, 4

Tags: algebra , system of equations , algebra unsolved

Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}\equal{}0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988$ holds for all $ i\equal{}1,...,1988$. Find all real solutions of the system of equations: $ \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988$.

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics

The sides of an equilateral triangle are divided into n equal parts by $n-1$ points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into $n^2$ equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by $60^{\circ}$ or by $120^{\circ}$ . a) Prove that, if $n \geq 7$, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle. b) Determine all the values of $n \geq 1$ for which the beetles can move along the sides of the small triangles without meeting in their vertices.

2019 Romania National Olympiad, 2

Tags: geometry , Squares , sqauare , equal segments , perpendicular

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.