Olympiad problems

1993 Tournament Of Towns, (358) 1

Tags: geometry , orthocenter

Let $M$ be a point on the side $AB$ of triangle $ABC$. The length $AB = c$ and $\angle CMA=\phi$ are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles $AMC$ and $BMC$. (IF Sharygin)

1988 IMO, 2

Tags: geometry , incenter , ratio , right triangle , area of a triangle

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

1972 IMO Longlists, 4

Tags: trigonometry , search , princeton , college

You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.

2023 Malaysian APMO Camp Selection Test, 1

Tags: algebra

For which $n\ge 3$ does there exist positive integers $a_1<a_2<\cdots <a_n$, such that: $$a_n=a_1+...+a_{n-1}, \hspace{0.5cm} \frac{1}{a_1}=\frac{1}{a_2}+...+\frac{1}{a_n}$$ are both true? [i]Proposed by Ivan Chan Kai Chin[/i]

1969 Bulgaria National Olympiad, Problem 4

Tags: triangle , geometry

Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.

2014 Purple Comet Problems, 14

Tags: probability , number theory , relatively prime

Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2011 National Chemistry Olympiad, 29

Tags:

Introduction of two drops of concentrated sulfuric acid, $\text{H}_2\text{SO}_4$, speeds up an esterification reaction. Introduction of a piece of platinum metal, $\text{Pt}$, speeds up the reaction of $\text{H}_2$ and $\text{O}_2$ gas. Which of the following statements is true? $ \textbf{(A)}\ \text{Pt is a homogeneous catalyst; sulfuric acid is a heterogeneous catalyst}\qquad$ $\textbf{(B)}\ \text{Pt is a heterogeneous catalyst; sulfuric acid is a homogeneous catalyst}\qquad$ $\textbf{(C)}\ \text{Pt and sulfuric acid are both heterogeneous catalysts}\qquad$ $\textbf{(D)}\ \text{Pt and sulfuric acid are both homogeneous catalysts}\qquad$

2004 Manhattan Mathematical Olympiad, 2

Tags:

Consider the sequence $1, \dfrac12 , \dfrac13 , \ldots$. Show that every positive rational number can be written as a finite sum of different terms in this sequence.

2010 Contests, 1

Tags: combinatorics , symmetry , subset , bounding

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

2015 Switzerland Team Selection Test, 1

Tags: combinatorics , chessboard

What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?

2003 Iran MO (3rd Round), 10

Tags: number theory

let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n find the number of natural divisors of na. :)

TNO 2008 Senior, 12

Tags: number theory

(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$. (b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $4n + 3$ is equal to the sum of the digits of $n$. (c) Prove that for any natural number $n$, it is possible to find $n$ consecutive numbers such that none of them is prime.

2008 Moldova Team Selection Test, 1

Tags: number theory

Let $ p$ be a prime number. Solve in $ \mathbb{N}_0\times\mathbb{N}_0$ the equation $ x^3\plus{}y^3\minus{}3xy\equal{}p\minus{}1$.

1990 Greece National Olympiad, 1

Tags: linear algebra , matrix

Let $A$ be a $2\,x\,2$ matrix with real numbers. Prove that if $A^3=\mathbb{O}$ then $A^2=\mathbb{O}$.

1996 Abels Math Contest (Norwegian MO), 2

Tags: algebra , floor function , function

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

2015 Canada National Olympiad, 3

Tags: combinatorics , square grid

On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

1984 Bundeswettbewerb Mathematik, 1

Tags: number theory , combinatorics , game , winning strategy

Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins. For which $n$ can $A$ and for which $n$ can $B$ force the win?

2023 Azerbaijan JBMO TST, 3

Tags: geometry

Let $ABC$ be a triangle and let $\Omega$ denote the circumcircle of $ABC$. The foot of altitude from $A$ to $BC$ is $D$. The foot of altitudes from $D$ to $AB$ and $AC$ are $K;L$ , respectively. Let $KL$ intersect $\Omega$ at $X;Y$, and let $AD$ intersect $\Omega$ at $Z$. Prove that $D$ is the incenter of triangle $XYZ$

2018 JHMT, 3

Tags: geometry

An equilateral triangle $ABC$ is in between two parallel lines $x, y$ that pass through points $A$ and $B$ respectively. Given that $C$ is twice as far from $y$ as $x$, the acute angle that $CA$ makes with $x$ is $\theta$. Then $(\tan \theta)^2$ is of the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

1976 IMO Longlists, 27

Tags: ratio , geometry

In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that $P$ divides the segment $AB$ in the ratio $1 : k$, $Q$ divides the segment $BC$ in the ratio $1 : l$, and $R$ divides the segment $CA$ in the ratio $1 : m.$

2022 Purple Comet Problems, 14

Tags: number theory

Of the integers $a$, $b$, and $c$ that satisfy $0 < c < b < a$ and $$a^3 - b^3 - c^3 - abc + 1 = 2022,$$ let $c'$ be the least value of $ c$ appearing in any solution, let $a'$ be the least value of $a$ appearing in any solution with $c = c'$, and let $b'$ be the value of $b$ in the solution where $c = c'$ and $a = a'$. Find $a' + b' + c'$.

2022 JHMT HS, 8

Tags: perimeter , optimization , calculus

Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?

2018 Switzerland - Final Round, 2

Tags: number theory , inequalities , gcd , greatest common divisor

Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$ . Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

1995 IberoAmerican, 3

Tags: 3d geometry , sphere , analytic geometry , function , similar triangles , angle bisector , geometry

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

2007 Pre-Preparation Course Examination, 8

Tags: number theory

Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.