For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Let $ABC$ be an acute triangle and let $X$ be a variable point on $AC$. The incircle of $\triangle ABX$ touches $AX, BX$ at $K, P$, respectively. The incircle of $\triangle BCX$ touches $CX, BX$ at $L, Q$, respectively. Find the locus of $KP \cap LQ$.

In the $3$-dimensional coordinate space nd the distance from the point $(36, 36, 36)$ to the plane that passes through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$.

Let $f(x)$ be a differentiable function. Find the following limit value: \[\lim_{n\to\infty} \dbinom{n}{k}\left\{f\left(\frac{x}{n}\right)-f(0)\right\}^k.\] Especially, for $f(x)=(x-\alpha)(x-\beta)$ find the limit value above. 1956 Tokyo Institute of Technology entrance exam

Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.

Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.

Consider a flat field on which there exist a valley in the form of an infinite strip with arbitrary width $\omega$. There exist a polyhedron of diameter $d$(Diameter in a polyhedron is the maximum distance from the points on the polyhedron) is in one side and a pit of diameter $d$ on the other side of the valley. We want to roll the polyhedron and put it into the pit such that the polyhedron and the field always meet each other in one point at least while rolling (If the polyhedron and the field meet each other in one point at least then the polyhedron would not fall into the valley). For crossing over the bridge, we have built a rectangular bridge with a width of $\frac{d}{10}$ over the bridge. Prove that we can always put the polyhedron into the pit considering the mentioned conditions. (You will earn a good score if you prove the decision for $\omega = 0$).

Kait rolls a fair $6$-sided die until she rolls a $6$. If she rolls a $6$ on the $N$th roll, she then rolls the die $N$ more times. What is the probability that she rolls a $6$ during these next N times?

Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to (a) no less than $ 5$ faces? (b) exactly $5$ faces? (G. Galperin)

How many zeros does $ f(x) \equal{} \cos(\log(x)))$ have on the interval $ 0 < x < 1$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{infinitely many}$

Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]

There are $n$ guests at a party. Any two guests are either friends or not friends. Every guest is friends with exactly four of the other guests. Whenever a guest is not friends with two other guests, those two other guests cannot be friends with each other either. What are the possible values of $n$?

Let $n$ be a given positive integer. Find the smallest positive integer $k$ for which the following statement is true: for any given simple connected graph $G$ and minimal cuts $V_1, V_2,\ldots, V_n$, at most $k$ vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer $1\le i\le n$ such that $V_i$ separates the two vertices. A partition of the vertices of $G$ into two disjoint non-empty sets is called a [i]minimal cut[/i] if the number of edges crossing the partition is minimal. [i]Proposed by András Imolay, Budapest[/i]

What is the largest prime factor of $7999488$?

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

Sum of the roots in the range $\left(-\frac{\pi}{2},\frac{\pi}{2} \right)$ of the equation $\sin x\tan x=x^2$ is [list=1] [*] $\frac{\pi}{2}$ [*] 0 [*] 1 [*] None of these [/list]

Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have? $\textbf{(A) }9\qquad\textbf{(B) }27\qquad\textbf{(C) }45\qquad\textbf{(D) }63\qquad\textbf{(E) }81$

Let $a_1, a_2,\cdots
,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots
(a + a_n - 1)$.

Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$

Find the smallest value of the expression $$16 \cdot \frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}$$

Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying: i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$; ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$. Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$. [i]Xiong Bin[/i]

A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$.

Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?

Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that (i) no three numbers of $A$ form an arithmetic progression, (ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$?

Tim starts with a number $n$, then repeatedly flips a fair coin. If it lands heads he subtracts 1 from his number and if it lands tails he subtracts 2. Let $E_n$ be the expected number of flips Tim does before his number is zero or negative. Find the pair $(a,b)$ such that \[ \lim_{n \to \infty} (E_n-an-b) = 0. \]