Let $n\geqslant 2$ be a fixed positive integer. Determine the minimum value of the expression $$\frac{a_{a_1}}{a_1}+\frac{a_{a_2}}{a_2}+\dots +\frac{a_{a_n}}{a_n},$$ where $a_1, a_2, \dots, a_n$ are positive integers at most $n$. [i]Proposed by David Anghel[/i]

Every positive integer greater than $1000$ is colored in red or blue, such that the product of any two distinct red numbers is blue. Is it possible to happen that no two blue numbers have difference $1$?

Let $p,q, r, s$ be real numbers with $q \ne -1$ and $s \ne -1$. Prove that the quadratic equations $x^2 + px+q = 0$ and $x^2 +rx+s = 0$ have a common root, while their other roots are inverse of each other, if and only if $pr = (q+1)(s+1)$ and $p(q+1)s = r(s+1)q$. (A double root is counted twice.)

Let $ ABC$ be a triangle, and $ D$ be a point where incircle touches side $ BC$. $ M$ is midpoint of $ BC$, and $ K$ is a point on $ BC$ such that $ AK\perp BC$. Let $ D'$ be a point on $ BC$ such that $ \frac{D'M}{D'K}=\frac{DM}{DK}$. Define $ \omega_{a}$ to be circle with diameter $ DD'$. We define $ \omega_{B},\omega_{C}$ similarly. Prove that every two of these circles are tangent.

Every square on the infinite sheet of cross-lined paper contains some real number. Prove that some square contains a number that does not exceed at least four of eight neighbouring numbers.

Ike and Mike go into a sandwich shop with a total of $\$30.00$ to spend. Sandwiches cost $\$4.50$ each and soft drinks cost $\$1.00$ each. Ike and Mike plan to buy as many sandwiches as they can and use any remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy? $\textbf{(A) } 6 \qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

Consider all positive multiples of $77$ less than $1,000,000.$ What is the sum of all the odd digits that show up?

Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2 $ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$.

In $\triangle ABC$, the interior sides of which are mirrors, a laser is placed at point $A_1$ on side $BC$. A laser beam exits the point $A_1$, hits side $AC$ at point $B_1$, and then reflects off the side. (Because this is a laser beam, every time it hits a side, the angle of incidence is equal to the angle of reflection). It then hits side $AB$ at point $C_1$, then side $BC$ at point $A_2$, then side $AC$ again at point $B_2$, then side $AB$ again at point $C_2$, then side $BC$ again at point $A_3$, and finally, side $AC$ again at point $B_3$. (a) Prove that $\angle B_3A_3C = \angle B_1A_1C$. (b) Prove that such a laser exists if and only if all the angles in $\triangle ABC$ are less than $90^{\circ}$.

Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

Let $\gamma_1$ and $\gamma_2$ be two disjoint circles, with centers $O_1$ and $O_2$. One of their exterior tangents cuts $\gamma_1$ in $A_1$ and $\gamma_2$ in $A_2$. One of their common internal tangents cuts $\gamma_1$ in $B_1$ and $\gamma_2$ in $B_2$, and the other common internal tangent cuts $\gamma_1$ in $C_1$ and $\gamma_2$ int $C_2$. Let $B_1B_2$ and $C_1C_2$ intersect in $O$. $X$ is the point where $A_2O$ cuts $\gamma_1$ and $OX<OB_1$. Similarly, $Y$ is the point where $A_1O$ cuts $\gamma_2$ and $OY<OB_2$. The perpendicular in $X$ to $OX$ cuts $O_1B_1$ in $P$ and the perpendicular in $Y$ to $OY$ cuts $O_2C_2$ in $Q$. Prove that $PQ$ and $A_1A_2$ are parallel. [i]Proposed by Flavian Georgescu[/i]

Triangle $ABC$ has $\overline{AB} = 5, \overline{BC} = 4, \overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$ and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?

Solve the following equation ${{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x$

A rook is allowed to move one cell either horizontally or vertically. After $64$ moves the rook visited all cells of the $8 \times 8$ chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal. (A Shapovalov, R Sadykov)

$n$ points were chosen on a circle. Two players are playing the following game: On every move a point is chosen and it is connected with an edge to an adjacent point or with the center of the circle. The winner is the player, after whose move each point can be reached by any other (including the center) by moving on the constructed edges. Find who of the two players has a winning strategy.