Olympiad problems

2015 NIMO Summer Contest, 6

Tags: combinatorics , set

Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set \[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \] [i] Proposed by Evan Chen [/i]

2017 Ukrainian Geometry Olympiad, 2

Tags: geometry , angle chasing , perpendicularity

On the side $AC$ of a triangle $ABC$, let a $K$ be a point such that $AK = 2KC$ and $\angle ABK = 2 \angle KBC$. Let $F$ be the midpoint of $AC$, $L$ be the projection of $A$ on $BK$. Prove that $FL \bot BC$.

2022 Belarusian National Olympiad, 8.3

Tags: geometry

Inside a square $ABCD$ point $P$ is marked, and on the sides $AB$, $BC$, $CD$ and $DA$ points $K,L,M$ and $N$ are chosen respectively. Lines $KP,LP,MP$ and $NP$ intersect sides $CD,DA,AB$ and $BC$ at points $K_1, L_1, M_1$ and $N_1$ respectively. It turned out that $$\frac{KP}{PK_1}+\frac{LP}{PL_1}+\frac{MP}{PM_1}+\frac{NP}{PN_1}=4$$ Prove that $KP+LP+MP+NP=K_1P+L_1P+M_1P+N_1P$.

2019-IMOC, A2

Tags: algebra , polynomial

Given a real number $t\ge3$, suppose a polynomial $f\in\mathbb R[x]$ satisfies $$\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.$$Prove that $\deg f\ge n$.

CIME I 2018, 13

Tags:

Find the number of positive integers $n<2017$ such that $n^2+n^0+n^1+n^7$ is not divisible by the square of any prime. [i]Proposed by [b]illogical_21[/b][/i]

2005 Brazil Undergrad MO, 1

Tags: linear algebra , matrix , algebra , polynomial , function

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

2015 China Team Selection Test, 4

Tags: triangle , algebra , geometry

Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.

2022 Canada National Olympiad, 2

Tags: number theory

I think we are allowed to discuss since its after 24 hours How do you do this Prove that $d(1)+d(3)+..+d(2n-1)\leq d(2)+d(4)+...d(2n)$ which $d(x)$ is the divisor function

2007 Cuba MO, 2

Tags: number theory , perfect square

Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

2018 Peru Cono Sur TST, 10

Tags: game , combinatorics

Let $n$ be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves: $(i)\:$ Choose a number of the form $2^j$, with $j$ a non-negative integer, and place it in an empty square. $(ii)\:$ Choose two (not necessarily consecutive) squares containing the same number, say $2^j$. Replace the number in one of the squares with $2^{j+1}$ and erase the number in the other square. At the end of the game, one square contains the number $2^n$, while the other squares are empty. Determine, as a function of $n$, the maximum number of turns Alex can make.

2018 Ukraine Team Selection Test, 8

Tags: function , algebra

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

2023 Israel TST, P1

Tags: combinatorics , pentagon

A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called [b] successful[/b] if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the maximum possible number of successful triples.

2014 Online Math Open Problems, 15

Tags: probability , ceiling function , linear algebra , matrix , modular arithmetic

In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$. Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$. [i]Proposed by Ray Li[/i]

2023 Balkan MO, 2

Tags: geometry

In triangle $ABC$, the incircle touches sides $BC,CA,AB$ at $D,E,F$ respectively. Assume there exists a point $X$ on the line $EF$ such that \[\angle{XBC} = \angle{XCB} = 45^{\circ}.\] Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$. Prove that the line $MD$ passes through $E$ or $F$. United Kingdom

2024 Turkey Olympic Revenge, 4

Tags: radical axis , geometry

Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle. Proposed by [i]Mehmet Can Baştemir[/i]

2004 China Team Selection Test, 2

Tags: combinatorics , inequalities

Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

2006 India IMO Training Camp, 3

Tags: modular arithmetic , invariant , combinatorics

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

2020 Cono Sur Olympiad, 5

Tags: invariant , product , operation

There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have.

2008 Regional Olympiad of Mexico Center Zone, 3

Tags: coloring , combinatorics

Consider a $n \times n$ grid divided into $n ^ 2$ squares of $1 \times 1$. Each of the $(n + 1) ^ 2 $ vertices of the grid is colored red or blue. Find the number of coloring such that each unit square has two red and two blue vertices.

2020 Durer Math Competition Finals, 1

Tags: equal area , geometry

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

2018-2019 Fall SDPC, 4

Tags: function , algebra

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-f(y))+2f(xy)=x^2f(x)+f(y^2)$$ for all real numbers $x,y$.

2008 IMO Shortlist, 5

Tags: combinatorics , probabilistic method , counting

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

2008 Purple Comet Problems, 13

Tags: probability , number theory , relatively prime

If you roll six fair dice, let $\mathsf{ p}$ be the probability that exactly five different numbers appear on the upper faces of the six dice. If $\mathsf{p} = \frac{m}{n}$ where $ m $ and $n$ are relatively prime positive integers, find $m+n.$

2014 China National Olympiad, 1

Tags: induction , algebra , number theory , logarithm

Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove: For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$ ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.

2017 ASDAN Math Tournament, 5

Tags:

Compute the maximum value attained by $f(x)=x^{1/x^2}$.