Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$. $BE, CF$ are the heights of $\triangle ABC$, and $BE, CF$ intersect at $H$. Let $M$ be the midpoint of $AH$, and $K$ be the point on $EF$ such that $HK \perp EF$. A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$, $Q$, respectively. Show that the tangent line of $(CQE)$ at $E$, the tangent line of $(BPF)$ at $F$, and $MK$ concur.

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

For how many ordered pairs of positive integers $(a, b)$ such that $a \le 50$ is it true that $x^2 - ax + b$ has integer roots?

Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.

Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.

Let $ABC$ be an acute, non-isosceles triangle with $AD$,$BE$, $CF$ are altitudes and $d$ is the tangent line of the circumcircle of triangle $ABC$ at $A$. The line through $H$ and parallel to $EF$ cuts $DE$, $DF$ at $Q, P$ respectively. Prove that $d$ is tangent to the ex-circle respect to vertex $D$ of triangle $DPQ$.

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$

James the naked mole rat is hopping on the number line. He starts at $0$ and jumps exactly $2^{n}$ either forward or backward at random at time $n$ seconds, his first jump being at time $n = 0$. What is the expected number of jumps James takes before he is on a number that exceeds $8$?

The equation $ax^5 + bx^4 + c = 0$ has three distinct roots. Show that so does the equation $cx^5 +bx+a = 0$.

Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]

Suppose that $ ABCD$ is a convex quadrilateral. Let $ F \equal{} AB\cap CD$, $ E \equal{} AD\cap BC$ and $ T \equal{} AC\cap BD$. Suppose that $ A,B,T,E$ lie on a circle which intersects with $ EF$ at $ P$. Prove that if $ M$ is midpoint of $ AB$, then $ \angle APM \equal{} \angle BPT$.

In [i]Terra Brasilis[/i] there are $n$ houses where $n$ goblins live, each in a house. There are one-way routes such that: - each route joins two houses, - in each house exactly one route begins, - in each house exactly one route ends. If a route goes from house $A$ to house $B$, then we will say that house $B$ is next to house $A$. This relationship is not symmetric, that is: in this situation, not necessarily house $A$ is next to house $B$. Every day, from day $1$, each goblin leaves the house where he is and arrives at the next house. A legend of [i]Terra Brasilis[/i] says that when all the goblins return to the original position, the world will end. a) Show that the world will end. b) If $n = 98$, show that it is possible for elves to build and guide the routes so that the world does not end before $300,000$ years.

Prove that there are two numbers $u$ and $v$, between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)$

Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition \[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\] for all real numbers $x$ and $y$.

Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer. (Slovakia)

The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]

Let $ABCD$ be a convex quadrilateral, such that $AB = CD$, $\angle BCD = 2 \angle BAD$, $\angle ABC = 2 \angle ADC$ and $\angle BAD \neq \angle ADC$. Determine the measure of the angle between the diagonals $AC$ and $BD$.

The ratio of $3$ to the positive number $n$ is the same as the ratio of $n$ to $192.$ Find $n.$

[b]Problem Section #1 c) Find all pairs $(m, n)$ of non-negative integers for which $m^2+2.3^n=m(2^{n+1}-1).$

Consider a multiplicative function \( f \) from the positive integers to the unit disk centered at the origin, that is, \( f : \mathbb{Z}^+ \to D^2 \subseteq \mathbb{C} \) such that \( f(mn) = f(m)f(n) \). Prove that for every \( \epsilon > 0 \) and every integer \( k > 0 \), there exist \( k \) distinct positive integers \( a_1, a_2, \dots, a_k \) such that \( \text{gcd}(a_1, a_2, \dots, a_k) = k \) and \( d(f(a_i), f(a_j)) < \epsilon \) for all \( i, j = 1, \dots, k \).

Given a rectangle $ABCD$ with $AB = a$ and $BC = b$. Point $O$ is the intersection of the two diagonals. Extend the side of the $BA$ so that $AE = AO$, also extend the diagonal of $BD$ so that $BZ = BO.$ Assume that triangle $EZC$ is equilateral. Prove that (i) $b = a\sqrt3$ (ii) $EO$ is perpendicular to $ZD$

Let $S$ be a nonempty set of points of the plane. We say that $S$ determines the distance $d > 0$ if there are two points $A, B$ in $S$ such that $AB = d$. Assuming that $S$ does not contain $8$ collinear points and that it determines not more than $91$ distances, prove that $S$ has less than $2004$ elements.

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? [asy] unitsize(1.25cm); dotfactor = 10; pen shortdashed=linetype(new real[] {2.7,2.7}); for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { draw((i,0)--(i,6), grey); draw((0,j)--(6,j), grey); } } for (int i = 1; i <= 6; ++i) { draw((-0.1,i)--(0.1,i),linewidth(1.25)); draw((i,-0.1)--(i,0.1),linewidth(1.25)); label(string(5*i), (i,0), 2*S); label(string(i), (0, i), 2*W); } draw((0,0)--(0,6)--(6,6)--(6,0)--(0,0)--cycle,linewidth(1.25)); label(rotate(90) * "Distance (miles)", (-0.5,3), W); label("Time (minutes)", (3,-0.5), S); dot("Naomi", (2,6), 3*dir(305)); dot((6,6)); label("Maya", (4.45,3.5)); draw((0,0)--(1.15,1.3)--(1.55,1.3)--(3.15,3.2)--(3.65,3.2)--(5.2,5.2)--(5.4,5.2)--(6,6),linewidth(1.35)); draw((0,0)--(0.4,0.1)--(1.15,3.7)--(1.6,3.7)--(2,6),linewidth(1.35)+shortdashed); [/asy] $\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

Let $G$ be the centroid of triangle $ABC$. Through $G$ draw a line parallel to $BC$ and intersecting the sides $AB$ and $AC$ at $P$ and $Q$ respectively. Let $BQ$ intersect $GC$ at $E$ and $CP$ intersect $GB$ at $F$. If $D$ is midpoint of $BC$, prove that triangles $ABC$ and $DEF$ are similar