Olympiad problems

2000 Irish Math Olympiad, 2

In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that: $ R^2\equal{}\frac{(ab\plus{}cd)(ac\plus{}bd)(ad\plus{}bc)}{16Q^2}$. Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.

1992 Baltic Way, 11

Tags: function , continued fraction , algebra

Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions: (i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$, (ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$, (iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.

2001 Putnam, 4

Tags: geometry , vector , geometric transformation , ratio , analytic geometry , symmetry

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

2019 IMEO, 4

Tags: polynomial , prime number , algebra

Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

2021 AMC 10 Spring, 1

Tags:

How many integer values satisfy $|x|<3\pi$? $\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20$

2015 Greece JBMO TST, 1

Tags: algebra , inequalities , three variable inequality

If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;

2000 AMC 8, 5

Tags:

Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period? $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 8$

1966 IMO Shortlist, 52

Tags: geometry , combinatorics , area , combinatorial geometry , dissection

A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$

2011 China Western Mathematical Olympiad, 1

Tags: inequalities , algebra

Given that $0 < x,y < 1$, find the maximum value of $\frac{xy(1-x-y)}{(x+y)(1-x)(1-y)}$

2002 Italy TST, 3

Tags: function , algebra

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2020 CMIMC Combinatorics & Computer Science, 2

Tags: combinatorics , computer science

David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that exactly $5$ of his answers are True. What is the probability he answers at least $5$ questions correctly?

2009 Czech and Slovak Olympiad III A, 5

Tags: combinatorics

At every vertex $A_k(1\le k\le n)$ of a regular $n$-gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$.

2017 China Team Selection Test, 3

Tags: quadratic residue , combinatorics , set theory

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$; ($ii$)$2017|x_1+...+x_{100}$; ($iii$)$2017|x_1^2+...+x_{100}^2$.

JOM 2025, 4

Tags: number theory

For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$. [i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]

2006 Indonesia MO, 1

Tags: algebra

Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.

2017 Sharygin Geometry Olympiad, P9

Tags: geometry

Let $C_0$ be the midpoint of hypotenuse $AB$ of triangle $ABC$; $AA_1, BB_1$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_0I$ and $A_1B_1$ meet on the altitude from $C$. [i]Proposed by A.Zaslavsky[/i]

2021 Tuymaada Olympiad, 5

Tags:

In a $100\times 100$ table $110$ unit squares are marked. Is it always possible to rearrange rows and columns so that all the marked unit squares are above the main diagonal or on it?

2005 Singapore MO Open, 4

Tags: combinatorics

Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.

1955 Putnam, B5

Tags:

Given an infinite sequence of $0$'s and $1$'s and a fixed integer $k,$ suppose that there are no more than $k$ distinct blocks of $k$ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence $11011010101$ followed by alternating $0$'s and $1$'s indefinitely, which is periodic beginning with the fifth term.)

2017 Saudi Arabia BMO TST, 2

Tags: incenter , concurrency , geometry

Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.

2014 IPhOO, 2

Tags: rotation , ratio

An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? $ \textbf {(A) } \dfrac {7}{10} \qquad \textbf {(B) } \dfrac {49}{100} \qquad \textbf {(C) } 1 \qquad \textbf {(C) } \dfrac {100}{49} \qquad \textbf {(E) } \dfrac {10}{7} $ [i]Problem proposed by Ahaan Rungta[/i]

Russian TST 2014, P2

Tags: geometry , incenter

A circle centered at $O{}$ passes through the vertices $B{}$ and $C{}$ of the acute-angles triangle $ABC$ and intersects the sides $AC{}$ and $AB{}$ at $D{}$ and $E{}$ respectively. The segments $CE$ and $BD$ intersect at $U{}.$ The ray $OU$ intersects the circumcircle of $ABC$ at $P{}.$ Prove that the incenters of the triangles $PEC$ and $PBD$ coincide.

2020 Iranian Combinatorics Olympiad, 2

Tags: combinatorics , probability

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

2022 Bolivia IMO TST, P4

Tags: combinatorics , graph theory , greatest common divisor

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

2010 LMT, 8

Tags:

How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$