$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

Let's call a function $f:\mathbb R\to\mathbb R$[i] weakly periodic[/i] if it is continuous and $f(x+1)=f(f(x))+1$ for all $x\in\mathbb R$. a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb R$? b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb R$? [i]Proposed by: András Imolay, Budapest[/i]

Jason and Valerie agree to meet for game night, which runs from $4:00$ PM to $5:00$ PM. Jason and Valerie each choose a random time from $4:00$ PM to $5:00$ PM to show up. If Jason arrives first, he will wait $20$ minutes for Valerie before leaving. If Valerie arrives first, she will wait $10$ minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?

Given an isosceles triangle $ ABC$ with $ \angle A \equal{} 90^{\circ}$. Let $ M$ be the midpoint of $ AB$. The line passing through $ A$ and perpendicular to $ CM$ intersects the side $ BC$ at $ P$. Prove that $ \angle AMC \equal{} \angle BMP$.

Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF \sim \triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.

If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that $$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$ Mathematical Excalibur P322/Vol.14, no.2

Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ dene $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.

A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of а) the vertices of their greatest angles, b) their incenters.

The ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle, to the perimeter of an equilateral triangle inscribed in the circle is: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 1: 3 \qquad\textbf{(C)}\ 1: \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3}: 2 \qquad\textbf{(E)}\ 2: 3$

Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

All the chairs in a classroom are arranged in a square $n\times n$ array (in other words, $n$ columns and $n$ rows), and every chair is occupied by a student. The teacher decides to rearrange the students according to the following two rules: (a) Every student must move to a new chair. (b) A student can only move to an adjacent chair in the same row or to an adjacent chair in the same column. In other words, each student can move only one chair horizontally or vertically. (Note that the rules above allow two students in adjacent chairs to exchange places.) Show that this procedure can be done if $n$ is even, and cannot be done if $n$ is odd.

A country with $n$ cities has some two-way roads connecting certain pairs of cities. Someone notices that if the country is split into two parts in any way, then there would be at most $kn$ roads between the two parts (where $k$ is a fixed positive integer). What is the largest integer $m$ (in terms of $n$ and $k$) such that there is guaranteed to be a set of $m$ cities, no two of which are directly connected by a road?

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Henry starts a trip when the hands of the clock are together between $ 8$ a.m. and $ 9$ a.m. He arrives at his destination between $ 2$ p.m. and $ 3$ p.m. when the hands of the clock are exactly $ 180^\circ$ apart. The trip takes: $ \textbf{(A)}\ \text{6 hr.} \qquad \textbf{(B)}\ \text{6 hr. 43\minus{}7/11 min.} \qquad \textbf{(C)}\ \text{5 hr. 16\minus{}4/11 min.} \qquad \textbf{(D)}\ \text{6 hr. 30 min.} \qquad \textbf{(E)}\ \text{none of these}$

Find all the integers $x$ in $[20, 50]$ such that $6x + 5 \equiv -19 \mod 10,$ that is, $10$ divides $(6x + 15) + 19.$

Let $ a,b,c>0$ prove that:\[ \frac{a^{3}}{(a+b)^{3}}+\frac{b^{3}}{(b+c)^{3}}+\frac{c^{3}}{(c+a)^{3}}\geq \frac{3}{8} \] Good luck! :D

Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$. The two circles intersect at $A$ and $B$. $\ell$ is the circles' common external tangent that is closer to $B$, and it meets $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Let $C$ be the point on line $AB$ not equal to $A$ that is the same distance from $\ell$ as $A$ is. Given that $O_1O_2=15$, $AT_1=5$ and $AT_2=12$, find $AC^2+{T_1T_2}^2$. [i]Proposed by Zachary Perry[/i]

The $2016$ players in the Gensokyo Tennis Club are playing Up and Down the River. The players first randomly form $1008$ pairs, and each pair is assigned to a tennis court (The courts are numbered from $1$ to $1008$). Every day, the two players on the same court play a match against each other to determine a winner and a loser. For $2\le i\le 1008$, the winner on court $i$ will move to court $i-1$ the next day (and the winner on court $1$ does not move). Likewise, for $1\le j\le 1007$, the loser on court $j$ will move to court $j+1$ the next day (and the loser on court $1008$ does not move). On Day $1$, Reimu is playing on court $123$ and Marisa is playing on court $876$. Find the smallest positive integer value of $n$ for which it is possible that Reimu and Marisa play one another on Day $n$. [i]Proposed by Yannick Yao[/i]

Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. To decide who goes first in a game, they play “countdown”. Henry starts by saying ‘34’, with Iggy saying ‘33’. If they continue to count down in their circular order, who will eventually say ‘1’? $\textbf{(A)}\ \text{Fred} \qquad \textbf{(B)}\ \text{Gail} \qquad \textbf{(C)}\ \text{Henry} \qquad \textbf{(D)}\ \text{Iggy} \qquad \textbf{(E)}\ \text{Joan}$

For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$? [i]Evan Chen and Ankan Bhattacharya[/i]

Find all $f: R\rightarrow R$ such that (i) The set $\{\frac{f(x)}{x}| x\in R-\{0\}\}$ is finite (ii) $f(x-1-f(x)) = f(x)-1-x$ for all $x$

Let $a,b,c$ be positive real numbers. Prove that \[\left((3a^2+1)^2+2\left(1+\frac{3}{b}\right)^2\right)\left((3b^2+1)^2+2\left(1+\frac{3}{c}\right)^2\right)\left((3c^2+1)^2+2\left(1+\frac{3}{a}\right)^2\right)\geq 48^3\]