In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$. Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$. Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of area $T_1$, and lines $E_cF_b$, $D_bE_a$, $F_aD_c$ bound a triangle of area $T_2$. What is the smallest possible value of the ratio $T_2/T_1$?

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the a) orthocenter b) centroid c) incenter of the triangle.

Area of convex quadrilateral $ABCD$ is $1$. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than $\frac{1}{4}$.

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.

A town's population increased by $1,200$ people, and then this new population decreased by $11 \%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population? $ \textbf{(A)}\ 1,200 \qquad\textbf{(B)}\ 11,200 \qquad\textbf{(C)}\ 9,968 \qquad\textbf{(D)}\ 10,000 \qquad\textbf{(E)}\ \text{none of these} $

For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

Let $a_1,a_2,...,a_9$ be a sequence of numbers satisfying $0 < p \le a_i \le q$ for each $i = 1,2,..., 9$. Prove that $\frac{a_1}{a_9}+\frac{a_2}{a_8}+...+\frac{a_9}{a_1} \le 1 + \frac{4(p^2+q^2)}{pq}$

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

For every positive integer $ n$, we define $ n?$ as $ 1?\equal{}1$ and $ n?\equal{}\frac{n}{(n\minus{}1)?}$ for $ n \ge 2$. Prove that $ \sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$

Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

Let $(f_n) _{n\ge 0}$ be the sequence defined by$ f_0 = 0, f_1 = 1, f_{n + 2 }= f_{n + 1} + f_n$ for $n> 0$ (Fibonacci string) and let $t_n =$ ${n+1}\choose{2}$ for $n \ge 1$ . Prove that: a) $f_1^2+f_2^2+...+f_n^2 = f_n \cdot f_{n+1}$ for $n \ge 1$ b) $\frac{1}{n^2} \cdot \Sigma_{k=1}^{n}\left( \frac{t_k}{f_k}\right)^2 \ge \frac{t_{n+1}^2}{9 f_n \cdot f_{n+1}}$

Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.

Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].

Solve the equation $$(2^x-4)^3 +(4^x-2)^3=(4^x+2^x-6)^3$$ where $x$ is a real number.

Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.

Find all natural numbers $m$ having exactly three prime divisors $p,q,r$, such that $$p-1\mid m; \quad qr-1 \mid m; \quad q-1 \nmid m; \quad r-1 \nmid m; \quad 3 \nmid q+r.$$

Determine the number of positive odd and even factor of $ 5^6\minus{}1$.

Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled? [i]Proposed by Andrew Wu[/i]

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

Consider the operation $*$ defined by the following table: \[\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}\] For example, $3*2=1$. Then $(2*4)*(1*3)=$ $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Can you find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition: a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$ b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$ for all $x,y,z$?