At a dinner party there are $N$ hosts and $N$ guests, seated around a circular table, where $N\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Prove that no matter how the $2N$ people are seated at the dinner party, at least $N$ pairs of guests will chat with one another.

Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.

Three coins are placed at the origin of a Cartesian coordinate system. On one move one removes a coin placed at some position $(x, y)$ and places three new coins at $(x+1, y)$, $(x, y+1)$ and $(x+1, y+1)$. Prove that after finitely many moves, there will exist two coins placed at the same point.

Consider the following figure formed by $10$ nodes and $15$ edges: [asy] unitsize(1.5 cm); pair A, B, C, D, E, F, G, H, I, J; A = dir(90); B = dir(90 + 360/5); C = dir(90 + 2*360/5); D = dir(90 + 3*360/5); E = dir(90 + 4*360/5); F = 0.6*A; G = 0.6*B; H = 0.6*C; I = 0.6*D; J = 0.6*E; draw(A--B--C--D--E--cycle); draw(F--H--J--G--I--cycle); draw(A--F); draw(B--G); draw(C--H); draw(D--I); draw(E--J); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(I); dot(J); [/asy] Prove that the edges of the figure cannot be colored by using $3$ different colors so that the edges that reach each node have different colors from each other.

Higher Secondary P1 A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.

Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$?

Eric and Christina are playing a game with $n$ stones. They alternate taking some number of stones from the pile, with Eric going first. The number of stones Eric takes from the pile must be a power of $3$ (e.g. 1, 3, 9, 27, ...), while the number of stones Christina takes must be a power of $2$ (e.g. 1, 2, 4, 8, ...). Whoever takes the last stone wins. Find the sum of all $1\leq n \leq 100$ for which Eric has a winning strategy. [i]Proposed by Connor Gordon[/i]

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Given a $101$-digit number $a$ and an arbitrary positive integer $b$. Prove that there is at most a $102$-digit positive integer $c$ such that any number of the form $\overline{caaa \dots ab}$ is composite.

Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$, $$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that $$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$

Given is an acute triangle $ ABC$ and the points $ A_1,B_1,C_1$, that are the feet of its altitudes from $ A,B,C$ respectively. A circle passes through $ A_1$ and $ B_1$ and touches the smaller arc $ AB$ of the circumcircle of $ ABC$ in point $ C_2$. Points $ A_2$ and $ B_2$ are defined analogously. Prove that the lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ have a common point, which lies on the Euler line of $ ABC$.

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

The diagram below shows a $ 4\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors. [asy]unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j));[/asy]Define a [i]growing path[/i] to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.

Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?

Let $a,b,c,d$ be positive integers such that $a+b+c+d=99$. Find the maximum and minimum of product $abcd$

Let $n \ge 2$ be an integer. Let $S$ be a set of $n$ elements and let $A_i, 1 \le i \le m$, be distinct subsets of $S$ of size at least $2$ such that $A_i \cap A_j \ne \emptyset$, $A_i \cap A_k \ne \emptyset$, $A_j \cap A_k \ne \emptyset$ imply $A_i \cap A_j \cap A_k \ne \emptyset$. Show that $m \le 2^{n-1}$ -

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.

There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

There is a broken computer such that only three primitive data $c$, $1$ and $-1$ are reserved. Only allowed operation may take $u$ and $v$ and output $u \cdot v + v.$ At the beginning, $u,v \in \{c, 1, -1\}.$ After then, it can also take the value of the previous step (only one step back) besides $\{c, 1, -1\}$. Prove that for any polynomial $P_{n}(x) = a_0 \cdot x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ with integer coefficients, the value of $P_n(c)$ can be computed using this computer after only finite operation.

For each positive integer $n\ge2$, find a polynomial $P_n(x)$ with rational coefficients such that $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$. (Note that $\sqrt[n]2$ denotes the positive $n^\text{th}$ root of $2$.)

Joseph rolls a fair 6-sided dice repeatedly until he gets 3 of the same side in a row. What is the expected value of the number of times he rolls? [i]Lightning 3.1[/i]

At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. Call a person [i]junior[/i] if they are younger than all their friends, and [i]senior[/i] if they are older than all their friends. A person with no friends is both [i]junior[/i] and [i]senior[/i]. A sequence of pairwise distinct people $A_1, \dots, A_m$ is called [i]photogenic[/i] if: 1. $A_1$ is [i]junior[/i], 2. $A_m$ is [i]senior[/i], and 3. $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \leq i \leq m-1$. Let $k$ be a positive integer such that for every [i]photogenic[/i] sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends. [i]Proposed by Shantanu Nene[/i]

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]