Olympiad problems

2015 International Zhautykov Olympiad, 3

The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality \[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]

2020 Stanford Mathematics Tournament, 2

Tags: geometry

Let $\vartriangle ABC$ be a right triangle with $\angle ABC = 90^o$. Let the circle with diameter $BC$ intersect $AC$ at $D$. Let the tangent to this circle at $D$ intersect $AB$ at $E$. What is the value of $\frac{AE}{BE}$ ?

2009 Peru Iberoamerican Team Selection Test, P4

Tags: geometry

Let $ABC$ be a triangle such that $AB < BC$. Plot the height $BH$ with $H$ in $AC$. Let I be the incenter of triangle $ABC$ and $M$ the midpoint of $AC$. If line $MI$ intersects $BH$ at point $N$, prove that $BN < IM$.

2001 Czech-Polish-Slovak Match, 5

Tags: function , limit , algebra unsolved , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy \[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]

2016 SDMO (High School), 5

Tags: combinatorial geometry , combinatorics unsolved , combinatorics

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

2012 Romanian Master of Mathematics, 5

Tags: floor function , ceiling function , geometry , rectangle , combinatorics proposed , combinatorics , double counting

Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours. [i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]

2018 Azerbaijan BMO TST, 2

Tags: Perfect Squares , functional equation , functional equation in N , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2007 Moldova Team Selection Test, 4

Tags: geometry , rhombus , Putnam , combinatorics proposed , combinatorics

We are given $n$ distinct points in the plane. Consider the number $\tau(n)$ of segments of length 1 joining pairs of these points. Show that $\tau(n)\leq \frac{n^{2}}3$.

2020 USA EGMO Team Selection Test, 5

Tags: graph theory , graph cycles , spanning tree , combinatorics , TST

Let $G = (V, E)$ be a finite simple graph on $n$ vertices. An edge $e$ of $G$ is called a [i]bottleneck[/i] if one can partition $V$ into two disjoint sets $A$ and $B$ such that [list] [*] at most $100$ edges of $G$ have one endpoint in $A$ and one endpoint in $B$; and [*] the edge $e$ is one such edge (meaning the edge $e$ also has one endpoint in $A$ and one endpoint in $B$). [/list] Prove that at most $100n$ edges of $G$ are bottlenecks. [i]Proposed by Yang Liu[/i]

2010 Contests, 3

Tags: function , limit , algebra unsolved , algebra

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

1999 Turkey MO (2nd round), 2

Tags: geometry proposed , geometry

Problem-2: Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that $\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.

2016 PAMO, 2

Tags: combinatorics

We have a pile of $2016$ cards and a hat. We take out one card, put it in the hat and then divide the remaining cards into two arbitrary non empty piles. In the next step, we choose one of the two piles, we move one card from this pile to the hat and then divide this pile into two arbitrary non empty piles. This procedure is repeated several times : in the $k$-th step $(k>1)$ we move one card from one of the piles existing after the step $(k-1)$ to the hat and then divide this pile into two non empty piles. Is it possible that after some number of steps we get all piles containing three cards each?

2006 MOP Homework, 5

Tags: graph theory , combinatorics unsolved , combinatorics

Smallville is populated by unmarried men and women, some of which are acquainted. The two City Matchmakers know who is acquainted with whom. One day, one of the matchmakers claimed: "I can arrange it so that every red haired man will marry a woman with who he is acquainted." The other matchmaker claimed: "I can arrange it so that every blonde woman will marry a man with whom she is acquainted." An amateur mathematician overheard this conversation and said: "Then it can be arranged so that every red haired man will marry a woman with whom he is acquainted and at the same time very blonde woman will marry a man with who she is acquainted." Is the mathematician right?

2021 JHMT HS, 5

Tags: expected value , 2021

The average of all ten-digit base-ten positive integers $\underline{d_9} \ \underline{d_8} \ldots \underline{d_1} \ \underline{d_0}$ that satisfy the property $|d_i - i| \leq 1$ for all $i \in \{0, 1, \ldots, 9\}$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Compute the remainder when $p + q$ is divided by $10^6.$

2014 Singapore Senior Math Olympiad, 3

Tags: induction

Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk than the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red. Suppose the bottom level has 2048 disks of which 2014 are red. What is the colour of the disk at the top?

1966 AMC 12/AHSME, 25

Tags: function

If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\ldots$, and $F(1)=2$, then $F(101)$ equals: $\text{(A)} \ 49 \qquad \text{(B)} \ 50 \qquad \text{(C)} \ 51 \qquad \text{(D)} \ 52 \qquad \text{(E)} \ 53$

2014 German National Olympiad, 2

Tags: combinatorics , number theory , Divisibility , Sequence , formula

For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$

2016 JBMO Shortlist, 5

Tags: JBMO , number theory

Determine all four-digit numbers $\overline{abcd} $ such that $(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $:

2006 Federal Math Competition of S&M, Problem 2

Tags: Diophantine equation , number theory

Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that $$\frac1x+\frac1y=\frac1p-\frac1q.$$

1997 Abels Math Contest (Norwegian MO), 2b

Tags: isosceles , circle , Product , independent , geometry

Let $A,B,C$ be different points on a circle such that $AB = AC$. Point $E$ lies on the segment $BC$, and $D \ne A$ is the intersection point of the circle and line $AE$. Show that the product $AE \cdot AD$ is independent of the choice of $E$.

1968 AMC 12/AHSME, 16

Tags: AMC

If $x$ is such that $\dfrac{1}{x}<2$ and $\dfrac{1}{x}>-3$, then: $\textbf{(A)}\ -\dfrac{1}{3}<x<\dfrac{1}{2} \qquad \textbf{(B)}\ -\dfrac{1}{2}<x<3 \qquad \textbf{(C)}\ x>\dfrac{1}{2} \qquad\\ \textbf{(D)}\ x>\dfrac{1}{2}\text{ or }-\dfrac{1}{3}<x<0 \qquad \textbf{(E)}\ x>\dfrac{1}{2}\text{ or }x<-\dfrac{1}{3}$

2017 Korea Junior Math Olympiad, 5

Tags: number theory , KJMO

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

2000 Tournament Of Towns, 1

Tags: Sum , combinatorics , table , square table

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

1947 Putnam, B5

Tags: Putnam , Integer Polynomial , roots

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

2021 OMpD, 5

Tags: geometry , angles , circumcircle

Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.