Olympiad problems

1990 IMO Longlists, 43

Let $V$ be a finite set of points in three-dimensional space. Let $S_1, S_2, S_3$ be the sets consisting of the orthogonal projections of the points of $V$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $| V|^2 \leq | S1|\cdot|S2|\cdot |S3|$, where $| A|$ denotes the number of elements in the finite set $A.$

2010 Today's Calculation Of Integral, 651

Tags: calculus , integration , limit , floor function , calculus computations

Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

2021 Girls in Math at Yale, 3

Tags: Yale , college

Suppose that $a_1 = 1,$ $a_2 = 2$, and for any $n \ge 3$, $a_n = a_1 + a_2 + \cdots + a_{n-1}$. Find $\frac{a_{2021}}{a_{2020}}$. [i]Proposed by Andrew Wu[/i]

1985 All Soviet Union Mathematical Olympiad, 406

Tags: lines , combinatorial geometry , maximum

$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.

1991 AMC 12/AHSME, 17

Tags: AMC

A positive integer $N$ is a [i]palindrome[/i] if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following two properties: (a) It is a palindrome (b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome. How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $

2009 Greece JBMO TST, 3

Tags: algebra , rational , integration , radical

Given are the non zero natural numbers $a,b,c$ such that the number $\frac{a\sqrt2+b\sqrt3}{b\sqrt2+c\sqrt3}$ is rational. Prove that the number $\frac{a^2+b^2+c^2}{a+b+c}$ is an integer .

2019 South East Mathematical Olympiad, 6

Tags: geometry

In $\triangle ABC$, $AB>AC$, the bisectors of $\angle ABC, \angle ACB$ meet sides $AC,AB$ at $D,E$ respectively. The tangent at $A$ to the circumcircle of $\triangle ABC$ intersects $ED$ extended at $P$. Suppose that $AP=BC$. Prove that $BD\parallel CP$.

2015 All-Russian Olympiad, 6

Tags: combinatorics unsolved , combinatorics

A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. [i](S.Berlov,A.Magazinov)[/i]

1999 Tournament Of Towns, 5

Tags: combinatorics , Sequence , Digits

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

1979 IMO Longlists, 45

Tags: modular arithmetic , function , number theory unsolved , number theory

For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.

2021 Iranian Geometry Olympiad, 1

Tags: geometry , symmetry , reflection

With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries. [img]https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

2021 Princeton University Math Competition, 4

Tags: number theory

Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.

2009 QEDMO 6th, 3

Tags: concurrency , concurrent , geometry

Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and $ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point. Note: For three pairs of different points $X, Y$ and $Z$ we define the [i]Circle [/i] $XYZ$ as the circumcircle of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight line.

1996 India Regional Mathematical Olympiad, 7

Tags:

If $A$ is a fifty element subset of the set $1,2,\ldots 100$ such that no two numbers from $A$ add up to $100$, show that $A$ contains a square.

2022 HMNT, 10

Tags:

A real number $x$ is chosen uniformly at random from the interval $[0, 1000].$ Find the probability that $$\left\lfloor\frac{\lfloor \tfrac{x}{2.5}\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor.$$

2007 Balkan MO Shortlist, G2

Tags: trigonometry , geometry , geometric transformation , reflection , trapezoid , circumcircle , trig identities

Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.

2016 Hanoi Open Mathematics Competitions, 1

Tags: number theory , Digits

How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ : (A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.

1999 Moldova Team Selection Test, 12

Tags: number theory

Solve the equation in postive integers $$x^2+y^2+1998=1997x-1999y.$$

1995 Greece National Olympiad, 3

Tags: algebra unsolved , algebra

If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

2020 USEMO, 1

Tags: USEMO , USEMO 2020

Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?

2004 National Olympiad First Round, 7

Tags:

At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8 $

2024 Lusophon Mathematical Olympiad, 5

Tags: combinatorics

In a $9\times9$ board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column. One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side. How many ways are there to paint the board such that the squares $44$ and $49$ are both black?

2023 BMT, Tie 1

Tags: algebra

Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.

1989 All Soviet Union Mathematical Olympiad, 497

Tags: equal ratios , geometry , areas , geometric inequality

$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.

2018 Ecuador NMO (OMEC), 5

Tags: cyclic quadrilateral , Concyclic , equal segments

Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.