Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

Does exist polynoms of one variable that are irreducible over the field of integers, have degree $ 60 $ and have multiples of the form $ X^n-1? $ If so, how many of them?

Mientka Publishing Company prices its bestseller [i]Where's Walter?[/i] as follows: \[C(n) \equal{} \begin{cases} 12n, &\text{if } 1 \le n \le 24\\ 11n, &\text{if } 25 \le n \le 48\\ 10n, &\text{if } 49 \le n \end{cases}\] where $ n$ is the number of books ordered, and $ C(n)$ is the cost in dollars of $ n$ books. Notice that $ 25$ books cost less than $ 24$ books. For how many values of $ n$ is it cheaper to buy more than $ n$ books than to buy exactly $ n$ books? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX\cdot TY$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R}\longrightarrow \mathbb{R}$ so that the equality $$f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2$$ is true for any real numbers $x,y$.

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Let $ S $ be a set of $ n $ concentric circles in the plane. Prove that if a function $ f: S\to S $ satisfies the property \[ d( f(A),f(B)) \geq d(A,B) \] for all $ A,B \in S $, then $ d(f(A),f(B)) = d(A,B) $, where $ d $ is the euclidean distance function.

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 $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.