Olympiad problems

2001 IMO Shortlist, 1

Tags: number theory , decimal representation , Digits , factorial , IMO Shortlist

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2007 China Team Selection Test, 3

Tags: least common multiple , number theory , relatively prime , number theory unsolved

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

1990 Brazil National Olympiad, 1

Tags: combinatorics proposed , combinatorics , Brazilian Math Olympiad , Brazilian Math Olympiad 1990

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

2024 CMIMC Algebra and Number Theory, 4

Tags: number theory

For positive integer $n$, let $f(n)$ be the largest integer $k$ such that $k!\leq n$, let $g(n)=n-(f(n))!$, and for $j\geq 1$ let $$g^j(n)=\underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.$$ Find the smallest positive integer $n$ such that $g^{j}(n)> 0$ for all $j<30$ and $g^{30}(n)=0$. [i]Proposed by Connor Gordon[/i]

Ukrainian TYM Qualifying - geometry, 2011.5

Tags: geometry , Locus , Circumcenter , fixed , Ukrainian TYM

The circle $\omega_0$ touches the line at point A. Let $R$ be a given positive number. We consider various circles $\omega$ of radius $R$ that touch a line $\ell$ and have two different points in common with the circle $\omega_0$. Let $D$ be the touchpoint of the circle $\omega_0$ with the line $\ell$, and the points of intersection of the circles $\omega$ and $\omega_0$ are denoted by $B$ and $C$ (Assume that the distance from point $B$ to the line $\ell$ is greater than the distance from point $C$ to this line). Find the locus of the centers of the circumscribed circles of all such triangles $ABD$.

1988 AMC 12/AHSME, 23

Tags: inequalities , geometry , 3D geometry , tetrahedron , AMC

The six edges of a tetrahedron $ABCD$ measure $7$, $13$, $18$, $27$, $36$ and $41$ units. If the length of edge $AB$ is $41$, then the length of edge $CD$ is $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 36 $

2005 India IMO Training Camp, 1

Tags: geometry solved , geometry

Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

2010 IMO Shortlist, 6

Tags: function , algebra , functional equation , IMO Shortlist , positive integers , Hi

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$ [i]Proposed by Alex Schreiber, Germany[/i]

2015 ISI Entrance Examination, 8

Tags: function , algebra , functional equation

Find all the functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$|f(x)-f(y)| = 2 |x - y| $$

2009 Tournament Of Towns, 3

Tags: symmetry , number theory proposed , number theory

Find all positive integers $a$ and $b$ such that $(a + b^2)(b + a^2) = 2^m$ for some integer $m.$ [i](6 points)[/i]

2000 Federal Competition For Advanced Students, Part 2, 2

Tags: geometry , trapezoid , geometry proposed

A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.

2020 IMC, 1

Tags: IMC , college contests , combinatorics , IMC 2020

Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties. 1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$ 2. $w$ contains an even number of $a$'s 3. $w$ contains an even number of $b$'s. For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$

2010 All-Russian Olympiad Regional Round, 11.7

Tags: algebra , trinomial

Integers $a,b,c$ are such that the values of the trinomials $bx^2+cx+a$ and $cx^2+ax+b$ at $x=1234$ coincide. Can the first trinomial at $x = 1$ take the value $2009$?

2009 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , Power , consecutive

A positive integer is called [i]saturated [/i]i f any prime factor occurs at a power greater than or equal to $2$ in its factorisation. For example, numbers $8 = 2^3$ and $9 = 3^2$ are saturated, moreover, they are consecutive. Prove that there exist infinitely many saturated consecutive numbers.

2014 AMC 12/AHSME, 4

Tags: AMC

Suppose that $a$ cows give $b$ gallons of milk in $c$ days. At this rate, how many gallons of milk will $d$ cows give in $e$ days? ${ \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$

2018 Azerbaijan IMO TST, 2

Tags: geometry , IMO Shortlist , geometry solved , homothety , tangent circles , power of a point , excircle

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 $\omega$.

2020-IMOC, A5

Tags: inequalities , IMOC

Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and \[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\] (From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

2019 Purple Comet Problems, 13

Tags: analytic geometry

There are relatively prime positive integers $m$ and $n$ so that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$ . Find $m + n$.

1970 AMC 12/AHSME, 20

Tags: AMC

Lines $HK$ and $BC$ lie in a plane. $M$ is the midpoint of line segment $BC$, and $BH$ and $CK$ are perpendicular to $HK$. Then we $\textbf{(A) }\text{always have }MH=MK\qquad\textbf{(B) }\text{always have }MH>BK\qquad$ $\textbf{(C) }\text{sometimes have }MH=MK\text{ but not always}\qquad$ $\textbf{(D) }\text{always have }MH>MB\qquad \textbf{(E) }\text{always have }BH<BC$

1975 Bulgaria National Olympiad, Problem 6

Tags: combinatorics , combinatorial geometry , geometry

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$. [i](H. Lesov)[/i]

2013 IberoAmerican, 6

Tags: rotation , geometry , geometric transformation , combinatorics proposed , combinatorics , combinatorial geometry

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

2004 Uzbekistan National Olympiad, 2

Tags: geometry , quadratics , number theory unsolved , number theory

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

2006 Taiwan TST Round 1, 2

Tags: analytic geometry , trigonometry , geometry , geometry proposed

Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.

2012 India PRMO, 5

Tags: number theory

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

2014 ELMO Shortlist, 9

Tags: logarithms , number theory proposed , number theory

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]