Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Find all triples $(p,n,k)$ of positive integers, where $p$ is a Fermat's Prime, satisfying \[p^n + n = (n+1)^k\]. [i]Observation: a Fermat's Prime is a prime number of the form $2^{\alpha} + 1$, for $\alpha$ positive integer.[/i]

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

A rectangle with area $n$ with $n$ positive integer, can be divided in $n$ squares(this squares are equal) and the rectangle also can be divided in $n + 98$ squares (the squares are equal). Find the sides of this rectangle

For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as \[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \] Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of \[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]

The positive real numbers $a,b,c,d$ satisfy the equality $$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$ Find the maximum possible value of $a$.

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.

Call a (non-trivial) lattice class a pseudo-variety if it is closed under taking a homomorphic image, a direct product, and a convex subset. Prove that the smallest distributive pseudo-variety cannot be defined by a first-order set of formulas.

A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?

Determine the value of $142857 + 285714 + 428571 + 571428.$ [i]Proposed by Ray Li[/i]

8. Let $\triangle A B C$ be a triangle with sidelengths $A B=5, B C=7$, and $C A=6$. Let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $L, M, N$ be the midpoints of sides $B C, C A, A B$, respectively. If the area of the convex hexagon with vertices at $D, E, F, L, M, N$ can be written as $\frac{x \sqrt{y}}{z}$ for positive integers $x, y, z$ with $\operatorname{gcd}(x, z)=1$ and $y$ square-free, find $x+y+z$.

$AB$ is a diameter of circle $O$. $X$ is a point on $AB$ such that $AX = 3BX.$ Distinct circles $\omega_1$ and $\omega_2$ are tangent to $O$ at $T_1$ and $T_2$ and to $AB$ at $X$. The lines $T_1X$ and $T_2X$ intersect $O$ again at $S_1$ and $S_2$. What is the ratio $\frac{T_1T_2}{S_1S_2}$?

Let $n \ge 3$ be an integer. A [i]circle dance[/i] is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle. At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow. a) Show that each circle dance visits each point infinitely often. b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.) [i](Birgit Vera Schmidt)[/i]

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

Prove, that for any positive real numbers $a, b, c$ who satisfy $a^2+b^2+c^2=1$ the following inequality holds. $\sqrt{\frac{1}{a}-a}+\sqrt{\frac{1}{b}-b}+\sqrt{\frac{1}{c}-c} \geq \sqrt{2a}+\sqrt{2b}+\sqrt{2c}$

Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\tfrac{AW}{BW} = \tfrac29$ , $\tfrac{AX}{CX} = \tfrac56$ , and $\tfrac{DY}{EY} = \tfrac92$. The ratio $\tfrac{EZ}{FZ}$ can then be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] size(200); defaultpen(linewidth(0.6)); real r = 3/11, s = 0.52, l = 33, d=5.5; pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0); pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F); draw(E--D--F--B--A--C^^W--Z); dot("$A$",A,N); dot("$B$",B,S); dot("$C$",C,S); dot("$D$",D,N); dot("$E$",E,S); dot("$F$",F,S); dot("$W$",W,0.6*NW); dot("$X$",X,0.8*NE); dot("$Y$",Y,dir(100)); dot("$Z$",Z,dir(70)); [/asy]

Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.

In $\Delta ABC$, $\angle ABC=120^\circ$. The internal bisector of $\angle B$ meets $AC$ at $D$. If $BD=1$, find the smallest possible value of $4BC+AB$.

Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.

The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square. [DIAGRAM NEEDED]

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

Given $\triangle ABC$, whose all sides have different length. Point $P$ is chosen on altitude $AD$. Lines $BP$ and $CP$ intersect lines $AC, AB$ respectively and point $X, Y$.It is given that $AX=AY$. Prove that there is circle, whose centre lies on $BC$ and is tangent to sides $AC$ and $AB$ at points $X,Y$.

Let $n$ be a fixed positive integer. Find the minimum value of $$\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}$$ where $x_1,x_2,\dots,x_n$ are distinct positive integers.

Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$. Points $A$ and $P$ lie on the same side of line $BC$. Point $R$ is an orthogonal projection of point $P$ on line $AC$. Point $S$ is middle of line segment $AQ$. Show that points $A, B, R, S$ lie on one circle.

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.