Given a triangle $\vartriangle ABC$,construct squares $BAQP$ and $ACRS$ outside the triangle $ABC$ (with vertices in that listed in counterclockwise order).Show that the line from $A$ perpendicular to $BC$ passes through the midpoint of the segment $QS$.

In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations $1,2,2^2,2^3,..$ Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for $2$ Bens in two ways : $2$ one Ben notes or $1$ two Ben note. For $5$ Ben one can given $1$ one Ben and $1$ four Ben note or $1$ Ben note and $2$ two Ben notes. Using $5$ one Ben notes or $3$ one Ben notes and $1$ two Ben notes for a $5$ Ben transaction is prohibited. Find the number of ways in which one can give a change $100$ Bens following the rules of the Government.

Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.

What is the least positive integer $k$ satisfying that $n+k\in S$ for every $n\in S$ where $S=\{n : n3^n + (2n+1)5^n \equiv 0 \pmod 7\}$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 42 $

Let $ABC$ be a scalene triangle and its incentre $I$. Denote by $F_A$ the intersection of the line $BC$ and the perpendicular to the angle bisector at $A$ through $I$. Let us define points $F_B$ and $F_C$ in a similar manner. Prove that points $F_A, F_B$ and $F_C$ are collinear.

There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and \[AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.\] Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

How many real numbers $a$ are there for which both solutions to the equation \[x^2 + (a - 2024)x + a = 0\] are integers? \[\mathrm a. ~15\qquad \mathrm b. ~16 \qquad \mathrm c. ~18 \qquad\mathrm d. ~20\qquad\mathrm e. ~24\qquad\]

For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational. [list](a) Prove the claim for $r \ge \frac43$ and $r \le 0$. (b) Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold. [/list]

In triangle $ABC$ we inscribe a square such that one of the sides of the square lies on the side $AC$, and the other two vertices lie on sides $AB$ and $BC$. Furthermore we know that $AC = 5$, $BC = 4$ and $AB = 3$. This square cuts out three smaller triangles from $\vartriangle ABC$. Express the sum of reciprocals of the inradii of these three small triangles as a fraction $p/q$ in lowest terms (i.e. with $p$ and $q$ coprime). What is $p + q$?

Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

Let $P$ be the power set of $\{1, 2, 3, 4\}$ (meaning the elements of P are the subsets of $\{1, 2, 3, 4\}$). How many subsets $S$ of $P$ are there such that no two distinct integers $a, b \in \{1, 2, 3, 4\}$ appear together in exactly one element of $S$?

Equilateral triangle $ABC$ has side length $20$. Let $PQRS$ be a square such that $A$ is the midpoint of $\overline{RS}$ and $Q$ is the midpoint of $\overline{BC}$. Compute the area of $PQRS$.

Find the $n$-th derivative with respect to $x$ of $$\int_{0}^{x} \left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{2!}+\ldots+\frac{(x-t)^{n-1}}{(n-1)!}\right)e^{nt} dt.$$

Suppose that the equation $x^3$+p$x^2$+qx+1 = 0; with p; q are rational numbers, has 3 real roots $x_1$; $x_2$; $x_3$; where $x_3 = 2 +\sqrt{5}$; compute the values of p and q?

I call two people $A$ and $B$ and think of a natural number $n$. Then I give the number $n$ to $A$ and the number $n+1$ to $B$. I tell them that they have both been given natural numbers, and further that they are consecutive natural numbers. However, I don't tell $A$ what $B$'s number is and vice versa. I start by asing $A$ if he knows $B$'s number. He says "no", Then I ask $B$ if he knows $A$'s number, and he says "no" too. I go back to $A$ and ask, and so on. $A$ and $B$ can both hear each other's responses. Do I ever get a "yes" in response? If so, who responds first with "yes" and how many times does he say "no" before this? Assume that both $A$ and $B$ are very intelligent and logical. You may need to consider multiple cases.

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

A chemist has $m$ ounces of salt that is $m\%$ salt. How many ounces of salt must he add to make a solution that is $2m\%$ salt? $ \textbf{(A)}\ \frac{m}{100+m} \qquad\textbf{(B)}\ \frac{2m}{100-2m}\qquad\textbf{(C)}\ \frac{m^2}{100-2m}\qquad\textbf{(D)}\ \frac{m^2}{100+2m}\qquad\textbf{(E)}\ \frac{2m}{100+2m} $

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.

Let $ABCD$ be a convex quadrilateral with $AB=CD$. Let $R,S$ be midpoints of sides $AD,BC$ respectively. Consider rays $AU, DV$ parallel with ray $RS$ and all of them point in the same direction. Show that $\angle BAU=\angle CDV$.

Let $ABCD$ be a square. $E$ and $F$ lie on sides $AB$ and $BC$, respectively, such that $BE = BF$. The line perpendicular to $CE$, which passes through $B$, intersects $CE$ and $AD$ at points $G$ and $H$, respectively. The lines $FH$ and $CE$ intersect at point $P$ and the lines $GF$ and $CD$ intersect at point $Q$. Prove that the line $DP$ is perpendicular to the line $BQ$.

In the coordinate plane, Yifan the Yak starts at $(0,0)$ and makes $11$ moves. In a move, Yifan can either do nothing or move from an arbitrary point $(i,j)$ to $(i+1,j)$, $(i,j+1)$ or $(i+1,j+1)$. How many points $(x,y)$ with integer coordinates exist such that the number of ways Yifan can end on $(x,y)$ is odd? [i]Proposed by Yifan Kang[/i]

Find all couples $(x,y)$ over the positive integers such that: $7^x+x^4+47=y^2$

Let points $A$, $B$, $C$ lie on a circle, and line $b$ be the tangent to the circle at point $B$. Perpendiculars $PA_1$ and $PC_1$ are dropped from a point $P$ on line $b$ onto lines $AB$ and $BC$ respectively. Points $A_1$ and $C_1$ lie inside line segments $AB$ and $BC$ respectively. Prove that $A_1C_1$ is perpendicular to $AC$.