Olympiad problems

1998 Singapore Team Selection Test, 1

The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that: \[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}. \]

2012 Mathcenter Contest + Longlist, 4 sl12

Tags: number theory

Given a natural $n>2$, let $\{ a_1,a_2,...,a_{\phi (n)} \} \subset \mathbb{Z}$ is the Reduced Residue System (RRS) set of modulo $n$ (also known as the set of integers $k$ where $(k,n)=1$ and no pairs are congruent in modulo $n$ ). if write $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{\phi (n)}}=\frac{a}{b}$$ where $a,b \in \mathbb{N}$ and $(a,b)=1$ , then prove that $n|a$. [i](PP-nine)[/i]

MOAA Team Rounds, 2022.15

Tags: geometry

Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, respectively. Let $O$ be the circumcenter of $ABC$. If $BI_B$ is perpendicular to $AO$, $AI_C = 3$ and $AC = 4\sqrt2$, then $AB^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Note: In triangle $\vartriangle ABC$, the $A$-excenter is the intersection of the exterior angle bisectors of $\angle ABC$ and $\angle ACB$. The $B$-excenter and $C$-excenter are defined similarly.

2000 Baltic Way, 9

Tags: combinatorics unsolved , combinatorics

There is a frog jumping on a $ 2k \times 2k$ chessboard, composed of unit squares. The frog's jumps are $ \sqrt{1 \plus{} k^2}$ long and they carry the frog from the center of a square to the center of another square. Some $ m$ squares of the board are marked with an $ \times$, and all the squares into which the frog can jump from an $ \times$'d square (whether they carry an $ \times$ or not) are marked with an $ \circ$. There are $ n$ $ \circ$'d squares. Prove that $ n \ge m$.

1929 Eotvos Mathematical Competition, 1

Tags: number theory

In how many ways can the sum of 100 fillér be made up with coins of denominations l, 2, 10, 20 and 50 fillér?

2018 Greece JBMO TST, 4

Tags: number theory , TST

Find all positive integers $x,y,z$ with $z$ odd, which satisfy the equation: $$2018^x=100^y + 1918^z$$

2010 Contests, 2

Tags: geometry , 3D geometry , tetrahedron , circumcircle , incenter , geometry proposed

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

2019 CMIMC, 2

Tags: 2019 , number theory

For all positive integers $n$, let $f(n)$ return the smallest positive integer $k$ for which $\tfrac{n}{k}$ is not an integer. For example, $f(6) = 4$ because $1$, $2$, and $3$ all divide $6$ but $4$ does not. Determine the largest possible value of $f(n)$ as $n$ ranges over the set $\{1,2,\ldots, 3000\}$.

2007 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

Triangle $ABC$ has $\angle A=90^\circ$, side $BC=25$, $AB>AC$, and area $150$. Circle $\omega$ is inscribed in $ABC$, with $M$ its point of tangency on $AC$. Line $BM$ meets $\omega$ a second time at point $L$. Find the length of segment $BL$.

2019 Romanian Master of Mathematics Shortlist, G5

Tags: geometry , perimeter

A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Prove that the lines $PI, A_1C_1$, and $B_1D_1$ are concurrent. Ankan Bhattacharya, USA

1981 Miklós Schweitzer, 3

Tags: topology

Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable. [i]A. Hajnal, I. Juhasz[/i]

1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5

Tags: inequalities

Find the smallest positive integer $ u$ such that there exists only one positive integer $ a$ and satisfies the inequality \[ 20u < 19a < 21u \ \text{?} \]

2006 ISI B.Math Entrance Exam, 4

Tags: calculus , derivative , function , induction , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2012 HMNT, 4

Tags: combinatorics

If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value?

2001 Abels Math Contest (Norwegian MO), 4

Tags: combinatorics

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties

2015 IFYM, Sozopol, 6

Tags: geometry , concurrency

The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.

2010 IFYM, Sozopol, 2

Tags: inequalities , inequalities unsolved

If $a,b,c>0$ and $abc=3$,find the biggest value of: $\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$

2011 Harvard-MIT Mathematics Tournament, 3

Tags: HMMT , geometry

Let $ABCDEF$ be a regular hexagon of area $1$. Let $M$ be the midpoint of $DE$. Let $X$ be the intersection of $AC$ and $BM$, let $Y$ be the intersection of $BF$ and $AM$, and let $Z$ be the intersection of $AC$ and $BF$. If $[P]$ denotes the area of a polygon $P$ for any polygon $P$ in the plane, evaluate $[BXC] + [AYF] + [ABZ] - [MXZY]$.

2006 Mathematics for Its Sake, 1

Tags: geometry , area , pigeonhole principle , combinatorics

Let be the points $ K,L,M $ on the sides $ BC,CA,AB, $ respectively, of a triangle $ ABC. $ Show that at least one of the areas of the triangles $ MAL,KBM,LCK $ doesn't surpass a fourth of the area of $ ABC. $

2024 Ukraine National Mathematical Olympiad, Problem 8

Tags: number theory , polynomial

Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

2020 BMT Fall, 22

Tags: algebra , system of equations

Suppose that $x, y$, and $z$ are positive real numbers satisfying $$\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases}$$ Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

1994 Tuymaada Olympiad, 3

Tags: geometry , geometric inequality

Point $M$ lies inside triangle $ABC$. Prove that for any other point $N$ lying inside the triangle $ABC$, at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$.

2021 MOAA, 6

Tags: MOAA 2021 , team

Find the sum of all two-digit prime numbers whose digits are also both prime numbers. [i]Proposed by Nathan Xiong[/i]

1997 IberoAmerican, 2

Tags: geometry , incenter , circumcircle , inradius , trapezoid , geometric transformation , rotation

In a triangle $ABC$, it is drawn a circumference with center in the incenter $I$ and that meet twice each of the sides of the triangle: the segment $BC$ on $D$ and $P$ (where $D$ is nearer two $B$); the segment $CA$ on $E$ and $Q$ (where $E$ is nearer to $C$); and the segment $AB$ on $F$ and $R$ ( where $F$ is nearer to $A$). Let $S$ be the point of intersection of the diagonals of the quadrilateral $EQFR$. Let $T$ be the point of intersection of the diagonals of the quadrilateral $FRDP$. Let $U$ be the point of intersection of the diagonals of the quadrilateral $DPEQ$. Show that the circumcircle to the triangle $\triangle{FRT}$, $\triangle{DPU}$ and $\triangle{EQS}$ have a unique point in common.

2006 AMC 8, 7

Tags: geometry

Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius. $ \textbf{(A)}\ X, Y, Z \qquad \textbf{(B)}\ Z, X, Y \qquad \textbf{(C)}\ Y, X, Z \qquad \textbf{(D)}\ Z, Y, X \qquad \textbf{(E)}\ X, Z, Y$