Olympiad problems

2007 Tuymaada Olympiad, 3

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

2016 Harvard-MIT Mathematics Tournament, 1

Tags: HMMT

If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.

Kyiv City MO Seniors Round2 2010+ geometry, 2020.11.2

Tags: geometry , parallelogram , circumcircle , equal angles

A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$. (Anton Trygub)

2001 May Olympiad, 2

Tags: geometry , trapezoid , rectangle

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

2024 India National Olympiad, 3

Tags: number theory , modular arithmetic , Divisibility

Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$ are divisible by $p$. Prove that $p$ divides each of $a,b,c$. $\quad$ Proposed by Navilarekallu Tejaswi

1998 USAMO, 4

Tags: rectangle , floor function , induction , combinatorics proposed , combinatorics , Extremal combinatorics , Hi

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

2023 Iranian Geometry Olympiad, 5

Tags: geometry , combinatorics , combinatorial geometry , triangulation , convex polygon

A polygon is decomposed into triangles by drawing some non-intersecting interior diagonals in such a way that for every pair of triangles of the triangulation sharing a common side, the sum of the angles opposite to this common side is greater than $180^o$. a) Prove that this polygon is convex. b) Prove that the circumcircle of every triangle used in the decomposition contains the entire polygon. [i]Proposed by Morteza Saghafian - Iran[/i]

2009 Germany Team Selection Test, 3

Tags: inequalities , algebra , IMO Shortlist

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

1993 Bundeswettbewerb Mathematik, 3

Tags: number theory , Perfect Squares , Perfect Square

There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: $16$ and $81$; $1681 = 41^2$. Prove that there are infinitely many pairs of squares with these properties.

1977 IMO Longlists, 12

Tags: modular arithmetic , number theory , relatively prime , number theory proposed

Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

2016 Ukraine Team Selection Test, 1

Tags: combinatorics , game , polygon

Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called [i]holes[/i]. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy? [i]Proposed by Bohdan Rublov [/i]

2023 IFYM, Sozopol, 2

Tags: functional equation , algebra

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[ f(x) + f(y - 1) + f(f(y - f(x))) = 1 \] for all integers $x$ and $y$.

2014-2015 SDML (High School), 3

Tags: 2014-15 SDML Round 1 , SDML High School , SDML Middle School

At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes? $\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$

2015 Canadian Mathematical Olympiad Qualification, 4

Tags: geometry , cyclic quadrilateral

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.

2016 MMATHS, 1

Tags: geometry , MMATHS

Let unit blocks be unit squares in the coordinate plane with vertices at lattice points (points $(a, b)$ such that $a$ and $b$ are both integers). Prove that a circle with area $\pi$ can cover parts of no more than $9$ unit blocks. The circle below covers part of $8$ unit blocks. [img]https://cdn.artofproblemsolving.com/attachments/4/4/43da9abed06d0feba94012ba68c177e3c2835b.png[/img]

2006 Hong Kong TST., 6

Tags: induction

Find $2^{2006}$ positive integers satisfying the following conditions. (i) Each positive integer has $2^{2005}$ digits. (ii) Each positive integer only has 7 or 8 in its digits. (iii) Among any two chosen integers, at most half of their corresponding digits are the same.

1996 French Mathematical Olympiad, Problem 2

Tags: algebra , Sequence

Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise. (a) Prove that one can find an element of $u_n$ smaller than $a$. (b) Prove that the sequence is eventually periodic.

2009 Math Prize For Girls Problems, 11

Tags: arithmetic sequence

An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?

2010 Contests, 3

Tags:

Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .

1956 AMC 12/AHSME, 2

Tags:

Mr. Jones sold two pipes at $ \$ 1.20$ each. Based on the cost, his profit one was $ 20 \%$ and his loss on the other was $ 20 \%$. On the sale of the pipes, he: $ \textbf{(A)}\ \text{broke even} \qquad\textbf{(B)}\ \text{lost } 4\text{ cents} \qquad\textbf{(C)}\ \text{gained } 4\text{ cents} \qquad\textbf{(D)}\ \text{lost } 10 \text{ cents} \qquad\textbf{(E)}\ \text{gained } 10 \text{ cents}$

1996 Yugoslav Team Selection Test, Problem 1

Tags: Sets , number theory , set , combinatorics , combinatorics unsolved

Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

2012 All-Russian Olympiad, 2

Tags: modular arithmetic , number theory proposed , number theory

Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?

2020 Kürschák Competition, P1

Tags: combinatorics , combinatorical geometry

Let $n$ and $k$ be positive integers. Given $n$ closed discs in the plane such that no matter how we choose $k + 1$ of them, there are always two of the chosen discs that have no common point. Prove that the $n$ discs can be partitioned into at most $10k$ classes such that any two discs in the same class have no common point.

V Soros Olympiad 1998 - 99 (Russia), 9.3

Tags: algebra , system of equations

Solve the system of equations: $$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$

Estonia Open Senior - geometry, 2017.2.5

Tags: geometry , diameter , angle bisector

The bisector of the exterior angle at vertex $C$ of the triangle $ABC$ intersects the bisector of the interior angle at vertex $B$ in point $K$. Consider the diameter of the circumcircle of the triangle $BCK$ whose one endpoint is $K$. Prove that $A$ lies on this diameter.