Olympiad problems

2012 NIMO Problems, 5

In $\triangle ABC$, $AB = 30$, $BC = 40$, and $CA = 50$. Squares $A_1A_2BC$, $B_1B_2AC$, and $C_1C_2AB$ are erected outside $\triangle ABC$, and the pairwise intersections of lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ are $P$, $Q$, and $R$. Compute the length of the shortest altitude of $\triangle PQR$. [i]Proposed by Lewis Chen[/i]

OIFMAT III 2013, 6

Tags: geometry , equal segments

The acute triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ be the intersection of the bisector of angle $BAC$ with segment $BC$ and $ P$ the intersection point of $AB$ with the perpendicular on $OA$ passing through $D$. Show that $AC = AP$.

2018 PUMaC Team Round, 16

Tags: PuMAC , Team Round , abstract algebra

Let $N$ be the number of subsets $B$ of the set $\{1,2,\dots,2018\}$ such that the sum of the elements of $B$ is congruent to $2018$ modulo $2048$. Find the remainder when $N$ is divided by $1000$.

2018 Iran MO (3rd Round), 1

Tags:

Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.

2005 Oral Moscow Geometry Olympiad, 4

Tags: geometry , parallelogram , pyramid , sphere

A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal. (M. Volchkevich)

2007 Today's Calculation Of Integral, 185

Tags: calculus , integration , trigonometry , calculus computations

Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

2017 Latvia Baltic Way TST, 14

Tags: number theory , Diophantine Equations , diophantine

Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

2021 Israel TST, 1

Tags: sum of digits , TST , number theory

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

MOAA Team Rounds, 2022.13

Tags: floor function , algebra

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

1983 IMO Longlists, 9

Tags: algebra , Sequence , Partial Orders , Divisibility , IMO Shortlist , combinatorics

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

2003 Tournament Of Towns, 1

Tags: combinatorics unsolved , combinatorics

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

2006 Brazil National Olympiad, 5

Tags: ratio , geometry unsolved , geometry

Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.

2015 Putnam, A4

Tags: Putnam , Putnam 2015

For each real number $x,$ let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x)\ge L$ for all $x\in [0,1)$? (As usual, $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$

1959 AMC 12/AHSME, 35

Tags: AMC

The symbol $\ge$ means "greater than or equal to"; the symbol $\le$ means "less than or equal to". In the equation $(x-m)^2-(x-n)^2=(m-n)^2$; m is a fixed positive number, and $n$ is a fixed negative number. The set of values $x$ satisfying the equation is: $ \textbf{(A)}\ x\ge 0 \qquad\textbf{(B)}\ x\le n\qquad\textbf{(C)}\ x=0\qquad\textbf{(D)}\ \text{the set of all real numbers}\qquad\textbf{(E)}\ \text{none of these} $

the 6th XMO, 2

Tags: complex numbers , algebra , inequalities , geometric inequality

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

2013 AMC 8, 4

Tags: AMC

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? $\textbf{(A)}\ \$120 \qquad \textbf{(B)}\ \$128 \qquad \textbf{(C)}\ \$140 \qquad \textbf{(D)}\ \$144 \qquad \textbf{(E)}\ \$160$

1940 Moscow Mathematical Olympiad, 066

Tags: combinatorial geometry , geometry , cone , 3D geometry

* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.

2007 Stanford Mathematics Tournament, 11

Tags: algebra , polynomial

The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.

2018 ELMO Shortlist, 3

Tags: inequalities

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]

1962 All Russian Mathematical Olympiad, 021

Tags: number theory

Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$.

1994 Mexico National Olympiad, 2

Tags: combinatorics , Integer , number theory

The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

1971 IMO Longlists, 45

Tags: combinatorics , length , point set , polygon , IMO Shortlist

A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

2001 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: functional equation , Find all functions , algebra

Find all functions $f: R \to R$ such that, for any $x, y \in R$: $f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$

2023 ELMO Shortlist, C7

Tags: Elmo , combinatorics

A [i]discrete hexagon with center $(a,b,c)$ \emph{(where $a$, $b$, $c$ are integers)[/i] and radius $r$ [i](a nonnegative integer)[/i]} is the set of lattice points $(x,y,z)$ such that $x+y+z=a+b+c$ and $\max(|x-a|,|y-b|,|z-c|)\le r$. Let $n$ be a nonnegative integer and $S$ be the set of triples $(x,y,z)$ of nonnegative integers such that $x+y+z=n$. If $S$ is partitioned into discrete hexagons, show that at least $n+1$ hexagons are needed. [i]Proposed by Linus Tang[/i]

III Soros Olympiad 1996 - 97 (Russia), 10.4

Tags: system of equations

Solve the system of equations $$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$