Olympiad problems

1982 Dutch Mathematical Olympiad, 1

Tags: inequalities

Which is greater: $ 17091982!^2$ or $ 17091982^{17091982}$?

2008 Princeton University Math Competition, B4

Tags: combinatorics

A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?

2006 Silk Road, 4

Tags: induction , geometry

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.

2014 Korea Junior Math Olympiad, 2

Tags: algebra , sum , inequalities , inequality

Let there be $2n$ positive reals $a_1,a_2,...,a_{2n}$. Let $s = a_1 + a_3 +...+ a_{2n-1}$, $t = a_2 + a_4 + ... + a_{2n}$, and $x_k = a_k + a_{k+1} + ... + a_{k+n-1}$ (indices are taken modulo $2n$). Prove that $$\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}$$

2023 India IMO Training Camp, 3

Tags: number theory , prime factorization , function , prime number , prime

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

1986 IMO Longlists, 14

Tags: rotation , fixed point , equilateral triangle , triangle , geometry

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

2009 AMC 8, 17

Tags: geometry , 3d geometry

The positive integers $ x$ and $ y$ are the two smallest positive integers for which the product of $ 360$ and $ x$ is a square and the product of $ 360$ and $ y$ is a cube. What is the sum of $ x$ and $ y$? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 85 \qquad \textbf{(C)}\ 115 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 610$

2021 AMC 12/AHSME Fall, 8

Tags: number theory , least common multiple

Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\frac{N}{M}?$ $(\textbf{A})\: 1\qquad(\textbf{B}) \: 2\qquad(\textbf{C}) \: 37\qquad(\textbf{D}) \: 74\qquad(\textbf{E}) \: 2886$

2016 Finnish National High School Mathematics Comp, 1

Tags: algebra , sidelengths , geometry

Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?

2007 Stanford Mathematics Tournament, 8

Tags: probability

Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?

2016 Harvard-MIT Mathematics Tournament, 7

Tags:

Let $q(x) = q^1(x) = 2x^2 + 2x - 1$, and let $q^n(x) = q(q^{n-1}(x))$ for $n > 1$. How many negative real roots does $q^{2016}(x)$ have?

2021 Austrian Junior Regional Competition, 3

Tags: combinatorics

The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

1979 Bundeswettbewerb Mathematik, 1

Tags: combinatorics

The plane is painted in red or blue color. Prove that you have a rectangle with the corners of the same color.

2005 AIME Problems, 15

Tags: geometry , quadratic , incenter , area of a triangle , heron's formula

Triangle $ABC$ has $BC=20$. The incircle of the triangle evenly trisects the median $AD$. If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n$.

2018 OMMock - Mexico National Olympiad Mock Exam, 1

Tags: geometry , trapezoid , circumcircle , parallel

Let $ABCD$ be a trapezoid with bases $AD$ and $BC$, and let $M$ be the midpoint of $CD$. The circumcircle of triangle $BCM$ meets $AC$ and $BD$ again at $E$ and $F$, with $E$ and $F$ distinct, and line $EF$ meets the circumcircle of triangle $AEM$ again at $P$. Prove that $CP$ is parallel to $BD$. [i]Proposed by Ariel García[/i]

2007 Bulgarian Autumn Math Competition, Problem 8.1

Tags: algebra , system of equations , parameter , parameterization

Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

2012 Spain Mathematical Olympiad, 3

Tags: circumcircle , geometry

Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.

2008 Indonesia TST, 2

Tags: combinatorics , subset

Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$

1990 IMO, 1

Tags: function , number theory , algebra , functional equation

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2010 AIME Problems, 3

Tags: factorial , number theory

Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

2022 Assara - South Russian Girl's MO, 4

Tags: number theory , greatest common divisor

Nadya has $2022$ cards, each with a number one or seven written on it. It is known that there are both cards.Nadya looked at all possible $2022$-digit numbers that can be composed from all these cards. What is the largest value that can take the greatest common divisor of all these numbers?

1983 AMC 12/AHSME, 16

Tags:

Let \[x = .123456789101112\ldots998999,\] where the digits are obtained by writing the integers 1 through 999 in order. The 1983rd digit to the right of the decimal point is $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

2003 Moldova Team Selection Test, 3

Tags: geometry

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

1968 Kurschak Competition, 2

Tags: combinatorial geometry , combinatorics

There are $4n$ segments of unit length inside a circle radius $n$. Show that given any line $L$ there is a chord of the circle parallel or perpendicular to $L$ which intersects at least two of the $4n$ segments.

1999 VJIMC, Problem 2

Tags: number theory

Find all natural numbers $n\ge1$ such that the implication $$(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)$$holds for any two natural numbers $a$ and $b$.