Found problems: 5802
Let $k\leq n$ be two positive integers. IMO-nation has $n$ villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite.
Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows $n$ and $k$. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than $k$. The office answers faithfully (infinite distance is larger than $k$). Prove that Alice can know whether the furthest two villages have finite distance between them in at most $2n^2/k$ calls.
[i]Proposed by usjl and Cheng-Ying Chang[/i]
For all odd natural numbers $ n, $ prove that
$$ \left|\sum_{j=0}^{n-1} (a+ib)^j\right|\in\mathbb{Q} , $$
where $ a,b\in\mathbb{Q} $ are two numbers such that $ 1=a^2+b^2. $
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Solve the equation $3^x=2^xy+1$ in positive integers.
Determine all sequences $ a_1,a_2,a_3,...$ of $ 1$ and $ \minus{}1$ such that $ a_{mn}\equal{}a_ma_n$ for all $ m,n$ and among any three successive terms $ a_n,a_{n\plus{}1},a_{n\plus{}2}$ both $ 1$ and $ \minus{}1$ occur.
Eve picked some apples, each weighing at most $\frac{1}{2}$ pound. Her apples weigh a total of $W$ pounds, where $W > \frac{1}{3}$. Prove that she can place all her apples into $\left\lceil \frac{3W - 1}{2} \right\rceil$ or fewer baskets, each of which holds up to 1 pound of apples. (The apples are not allowed to be cut into pieces.) Note: If $x$ is a real number, then $\lceil x \rceil$ (the ceiling of $x$) is the least integer that is greater than or equal to $x$.
Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).
Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs.
Juniors) How many vertical sides can there be in this set?
Seniors) How many ways are there to do that?
[asy]
size(120);
defaultpen(linewidth(0.8));
path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;
for(int i=0;i<=3;i=i+1)
{
for(int j=0;j<=2;j=j+1)
{
real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;
draw(shift(shiftx,shifty)*hex);
}
}
[/asy]
[i](T. Doslic)[/i]
Let $ \left( a_n \right) ,\left( b_n \right) $ be two sequences of real numbers from the interval $ (-1,1) $ having the property that
$$ \max\left( \left| a_{n+1} -a_n \right| ,\left| b_{n+1} -b_n \right| \right) \le\frac{1}{(n+4)(n+5)} , $$
for any natural number.
Prove that $ \left| a_nb_n -a_1b_1 \right|\le 1/2, $ for any natural number $ n. $
[i]Cristinel Mortici[/i]
Find all pairs of integers $(x,y)$ for which $x^z+z^x=(x+z)!$.
Let $ G=(V,E)$ be a simple graph.
a) Let $ A,B$ be a subsets of $ E$, and spanning subgraphs of $ G$ with edges $ A,B,A\cup B$ and $ A\cap B$ have $ a,b,c$ and $ d$ connected components respectively. Prove that $ a+b\leq c+d$.
We say that subsets $ A_1,A_2,\dots,A_m$ of $ E$ have $ (R)$ property if and only if for each $ I\subset\{1,2,\dots,m\}$ the spanning subgraph of $ G$ with edges $ \cup_{i\in I}A_i$ has at most $ n-|I|$ connected components.
b) Prove that when $ A_1,\dots,A_m,B$ have $ (R)$ property, and $ |B|\geq2$, there exists an $ x\in B$ such that $ A_1,A_2,\dots,A_m,B\backslash\{x\}$ also have property $ (R)$.
Suppose that edges of $ G$ are colored arbitrarily. A spanning subtree in $ G$ is called colorful if and only if it does not have any two edges with the same color.
c) Prove that $ G$ has a colorful subtree if and only if for each partition of $ V$ to $ k$ non-empty subsets such as $ V_1,\dots,V_k$, there are at least $ k\minus{}1$ edges with distinct colors that each of these edges has its two ends in two different $ V_i$s.
d) Assume that edges of $ K_n$ has been colored such that each color is repeated $ \left[\frac n2\right]$ times. Prove that there exists a colorful subtree.
e) Prove that in part d) if $ n\geq5$ there is a colorful subtree that is non-isomorphic to $ K_{1,n-1}$.
f) Prove that in part e) there are at least two non-intersecting colorful subtrees.
Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin $2013$ times. On the $n^{\text{th}}$ flip (where $n=1,2,\dots,2013$), Ben does the following if the coin flips heads:
(i) If the blackboard is empty, Ben writes $n$ on the blackboard.
(ii) If the blackboard is not empty, let $m$ denote the largest number on the blackboard. If $m^2+2n^2$ is divisible by $3$, Ben erases $m$ from the blackboard; otherwise, he writes the number $n$.
No action is taken when the coin flips tails. If probability that the blackboard is empty after all $2013$ flips is $\frac{2u+1}{2^k(2v+1)}$, where $u$, $v$, and $k$ are nonnegative integers, compute $k$.
[i]Proposed by Evan Chen[/i]
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Prove
\[ \frac{1}{\cos 0^\circ \cos 1^\circ} + \frac{1}{\cos 1^\circ \cos 2^\circ} + \cdots + \frac{1}{\cos 88^\circ \cos 89^\circ} = \frac{\cos 1^\circ}{\sin^2 1^\circ}. \]
Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying
$$P(x + Q(y)) = Q(x + P(y))$$
for all real numbers $x$ and $y$.
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened.
Decide whether there exists a strategy for player $A$ to win in a finite number of moves.
Given a positive integer $n$, let $[n] = \{1,2,\dots,n\}$. Let
[list]
[*] $a_n$ denote the number of functions $f: [n] \to [n]$ such that $f(f(i))\ge i$ for all $i$; and
[*] $b_n$ denote the number of ordered set partitions of $[n]$, i.e., the number of ways to pick an integer $k$ and an ordered $k$-tuple of pairwise disjoint nonempty sets $(A_1,\dots,A_k)$ whose union is $[n]$.
[/list]
Prove that $a_n=b_n$.
[i]Derek Liu[/i]
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and
\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]
for all $x, y \in\mathbb{R}$.
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.
There are $n$ natural numbers written on the board. Every move, we could erase $a,b$ and change it to $\gcd(a,b)$ and $\text{lcm}(a,b) - \gcd(a,b)$. Prove that in finite number of moves, all numbers in the board could be made to be equal.
Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that
(i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \]
(ii) For all $x \in [0,1)$, $f(0) \geq f(x)$,
(iii) $-f(-1) = f(1) = 1$.
Find all such functions $f$.
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.