Olympiad problems

1971 AMC 12/AHSME, 9

An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. If the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance between the centers of the pulleys in inches is $\textbf{(A) }24\qquad\textbf{(B) }2\sqrt{119}\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }4\sqrt{35}$

1988 Mexico National Olympiad, 2

Tags: number theory , divisor

If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .

2013-2014 SDML (High School), 12

Tags:

Compute the remainder when $20^{\left(13^{14}\right)}$ is divided by $11$. $\text{(A) }1\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }9$

Kvant 2019, M2561

Tags: combinatorics , phi function

On the grid plane all possible broken lines with the following properties are constructed: each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$. (Ilke Chanakchi, Ralf Schiffler)

2012 Purple Comet Problems, 23

Tags: modular arithmetic

Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.

2008 Sharygin Geometry Olympiad, 5

Tags: geometric transformation , rotation , geometry

(Kiev olympiad, 8--9) Reconstruct the square $ ABCD$, given its vertex $ A$ and distances of vertices $ B$ and $ D$ from a fixed point $ O$ in the plane.

1993 Tournament Of Towns, (371) 3

Tags: pascal triangle , algebra

Each number in the second, third, and further rows of the following triangle: [img]https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png[/img] is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle). (GW Leibnitz,)

2012 Switzerland - Final Round, 8

Tags: combinatorics , combinatorial geometry , 3d geometry , geometry

Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.

2022 Macedonian Mathematical Olympiad, Problem 3

Tags: prime divisor , sequence , number theory

The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$. For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$. [i]Proposed by Nikola Velov[/i]

2022 Stanford Mathematics Tournament, 7

Tags:

Let $M=\{0,1,2,\dots,2022\}$ and let $f:M\times M\to M$ such that for any $a,b\in M$, \[f(a,f(b,a))=b\] and $f(x,x)\neq x$ for each $x\in M$. How many possible functions $f$ are there $\pmod{1000}$?

2011 Belarus Team Selection Test, 4

Tags: combinatorics

Given a $n \times n$ square table. Exactly one beetle sits in each cell of the table. At $12.00$ all beetles creeps to some neighbouring cell (two cells are neighbouring if they have the common side). Find the greatest number of cells which can become empty (i.e. without beetles) if a) $n=8$ b) $n=9$ Problem Committee of BMO 2011

2016 Mathematical Talent Reward Programme, MCQ: P 8

Tags: prime number , number theory

Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is [list=1] [*] 283 [*] 307 [*] 593 [*] 691 [/list]

2007 Bulgaria National Olympiad, 3

Tags: function , trigonometry , polynomial , search , algebra

Find the least positive integer $n$ such that $\cos\frac{\pi}{n}$ cannot be written in the form $p+\sqrt{q}+\sqrt[3]{r}$ with $p,q,r\in\mathbb{Q}$. [i]O. Mushkarov, N. Nikolov[/i] [hide]No-one in the competition scored more than 2 points[/hide]

2004 All-Russian Olympiad, 4

Tags: algorithm , combinatorics

A rectangular array has 9 rows and 2004 columns. In the 9 * 2004 cells of the table we place the numbers from 1 to 2004, each 9 times. And we do this in such a way that two numbers, which stand in exactly the same column in and differ around at most by 3. Find the smallest possible sum of all numbers in the first row.

2022 Rioplatense Mathematical Olympiad, 6

Tags: combinatorics

Let $N(a,b)$ be the number of ways to cover a table $a \times b$ with domino tiles. Let $M(a,2b+1)$ be the number of ways to cover a table $a \times 2b+1$ with domino tiles, such that there are [b]no[/b] vertical tile in the central column. Prove that $$M(2m,2n+1)=2^m \cdot N(2m,n)\cdot N(2m,n-1)$$

2008 Baltic Way, 12

Tags: function , modular arithmetic , divisibility tests , graph theory , combinatorics , number theory

In a school class with $ 3n$ children, any two children make a common present to exactly one other child. Prove that for all odd $ n$ it is possible that the following holds: For any three children $ A$, $ B$ and $ C$ in the class, if $ A$ and $ B$ make a present to $ C$ then $ A$ and $ C$ make a present to $ B$.

2019 Middle European Mathematical Olympiad, 6

Tags: geometry , circumcircle

Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear. [i]Proposed by Dominik Burek, Poland[/i]

1998 Gauss, 10

Tags: gauss

At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by $0.25$ seconds. If Bonnie’s time was exactly $7.80$ seconds, how long did it take for Wendy to go down the slide? $\textbf{(A)}\ 7.80~ \text{seconds} \qquad \textbf{(B)}\ 8.05~ \text{seconds} \qquad \textbf{(C)}\ 7.55~ \text{seconds} \qquad \textbf{(D)}\ 7.15~ \text{seconds} \qquad $ $\textbf{(E)}\ 7.50~ \text{seconds}$

1968 IMO Shortlist, 21

Tags: number theory , divisibility

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

2012 JHMT, 5

Tags: geometry

Let $ABC$ be an equilateral triangle with side length $1$. Draw three circles $O_a$, $O_b$, and $O_c$ with diameters BC, $CA$, and $AB$, respectively. Let $S_a$ denote the area of the region inside $O_a$ and outside of $O_b$ and $O_c$. Define $S_b$ and $S_c$ similarly, and let $S$ be the area of the region inside all three circles. Find $S_a + S_b + S_c - S$.

2005 Germany Team Selection Test, 1

Tags: number theory

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

1996 Tuymaada Olympiad, 2

Tags: algebra , set theory , real , set

Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?

LMT Speed Rounds, 24

Tags: algebra

Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$ [i]Proposed by Evin Liang[/i]

1978 Romania Team Selection Test, 4

Tags: perimeter , rhombus , pure geometry , geometry

Diagonals $ AC $ and $ BD $ of a convex quadrilateral $ ABCD $ intersect a point $ O. $ Prove that if triangles $ OAB,OBC,OCD $ and $ ODA $ have the same perimeter, then $ ABCD $ is a rhombus. What happens if $ O $ is some other point inside the quadrilateral?

2014 NIMO Problems, 14

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

Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$. Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]