Olympiad problems

1993 AIME Problems, 14

Tags: geometry , rectangle , rotation , perimeter , geometric transformation , circumcircle , analytic geometry

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

1997 Putnam, 5

Tags:

Let $N_k$ denote number of ordered $n$-tuples of positive integers $(a_1,a_2, \cdots ,a_k)$ such that \[ \frac{1}{a_1}+\frac{1}{a_2}+\ldots +\frac{1}{a_k}=1 \] Determine $N_{10}$ is odd or even.

Indonesia MO Shortlist - geometry, g4

Tags: geometry , tangent circle

Given that two circles $\sigma_1$ and $\sigma_2$ internally tangent at $N$ so that $\sigma_2$ is inside $\sigma_1$. The points $Q$ and $R$ lies at $\sigma_1$ and $\sigma_2$, respectively, such that $N,R,Q$ are collinear. A line through $Q$ intersects $\sigma_2$ at $S$ and intersects $\sigma_1$ at $O$. The line through $N$ and $S$ intersects $\sigma_1$ at $P$. Prove that $$\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.$$

1985 Tournament Of Towns, (098) 2

Tags: game strategy , game , combinatorics

In the game "cat and mouse" the cat chases the mouse in either labyrinth $A, B$ or $C$ . [img]https://cdn.artofproblemsolving.com/attachments/4/5/429d106736946011f4607cf95956dcb0937c84.png[/img] The cat makes the first move starting at the point marked "$K$" , moving along a marked line to an adjacent point . The mouse then moves , under the same rules, starting from the point marked "$M$" . Then the cat moves again, and so on . If, at a point of time , the cat and mouse are at the same point the cat eats the mouse. Is there available to the cat a strategy which would enable it to catch the mouse , in cases $A, B$ and $C$? (A. Sosinskiy, Moscow)

2003 Manhattan Mathematical Olympiad, 2

Tags: pigeonhole principle

A tennis net is made of strings tied up together which make a grid consisting of small congruent squares as shown below. [asy] size(500); xaxis(-50,50); yaxis(-5,5); add(shift(-50,-5)*grid(100,10));[/asy] The size of the net is $100\times 10$ small squares. What is the maximal number of edges of small squares which can be cut without breaking the net into two pieces? (If an edge is cut, the cut is made in the middle, not at the ends.)

2018 BMT Spring, 1

Tags: combinatorics

Bob has $3$ different fountain pens and $11$ different ink colors. How many ways can he fill his fountain pens with ink if he can only put one ink in each pen?

2018 China Western Mathematical Olympiad, 7

Tags: number theory , least common multiple , inequalities

Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

2024 CMIMC Team, 5

Tags: team

An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die. Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.) [i]Proposed by Lohith Tummala[/i]

2017 IMO Shortlist, G4

Tags: homothety , power of a point , excircle , geometry , tangent circle

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

1935 Moscow Mathematical Olympiad, 018

Tags: sum , sum of cubes , algebra

Evaluate the sum: $1^3 + 3^3 + 5^3 +... + (2n - 1)^3$.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.1

Tags: number theory

The number $1995$ is divisible by both $19$ and $95$. How many four-digit numbers are there that are divisible by two-digit numbers formed by both its first two digits and its last two digits?

1997 AIME Problems, 4

Tags: number theory , relatively prime , pythagorean theorem , geometry

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

JBMO Geometry Collection, 2008

Tags: trigonometry , angle bisector , perpendicular bisector , power of a point , geometry

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

2022 Thailand TST, 3

Tags: number theory

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2018 HMNT, 5

Tags: algebra

Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.

1992 Czech And Slovak Olympiad IIIA, 4

Tags: trigonometry , algebra

Solve the equation $\cos 12x = 5\sin 3x+9\ tan ^2x+\ cot ^2x$

2012 All-Russian Olympiad, 1

Tags: combinatorics

$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.

2009 Tuymaada Olympiad, 2

Tags: trapezoid , geometric transformation , homothety , angle bisector , geometry

$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$. [i]Proposed by S. Berlov[/i]

Indonesia MO Shortlist - geometry, g8

Tags: integer , geometry , consecutive , equal angles

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

2005 Thailand Mathematical Olympiad, 6

Tags: number theory , diophantine equation , diophantine

Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

2019 Romanian Master of Mathematics Shortlist, original P6

Tags: polynomial , multivariate polynomial , algebra

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

2024 ITAMO, 2

Tags: locus , geometry

We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$. Determine the geometric locus of all median points.