Compute the number of positive integers $n\leq1330$ for which $\tbinom{2n}{n}$ is not divisible by $11$.

Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements

Thin and Fat eat a pizza of $2n$ pieces. Each piece contains a distinct amount of olives between $1$ and $2n$. Thin eats the first piece, and the two players alternately eat a piece neighbor of an eaten piece. However, neither Thin nor Fat like olives, so they will choose pieces that minimizes the total amount of olives they eat. For each arrangement $\sigma$ of the olives, let $s(\sigma)$ the minimal amount of olives that Thin can eat, considering that both play in the best way possible. Let $S(n)$ the maximum of $s(\sigma)$, considering all arrangements. $a)$ Prove that $n^2-1+\lfloor \frac{n}{2} \rfloor \le S(n) \le n^2+\lfloor \frac{n}{2} \rfloor$ $b)$ Prove that $S(n)=n^2-1+\frac{n}{2}$ for each even n.

On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.

Four heaps contain $38,45,61$ and $70$ matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?

Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? [asy] path p=origin--(18,0)--(18,5)--(0,5)--cycle; draw(p^^shift(0,8)*p^^shift(22,0)*p^^shift(22,16)*p^^shift(22,24)*p); filldraw(shift(0,16)*p^^shift(22,8)*p^^shift(22,32)*p^^shift(0,32)*p^^shift(0,24)*p, black, black); draw((-1,-1)--(41,-1)--(41,38)--(-1,38)--cycle, linewidth(2)); int i; for(i=1; i<6; i=i+1) { label(string(6-i), (-3,8*i-5.5), W); label(string(11-i), (43,8*i-5.5), E); }[/asy]

For a positive integer $n$, define the $n$th triangular number $T_n$ to be $\frac{n(n+1)}{2}$, and define the $n$th square number $S_n$ to be $n^2$. Find the value of \[\sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}.\] [i]Proposed by Yannick Yao[/i]

Define $M_n = \{ 1, 2, \ldots , n \} $ for all positive integers $n$. A collection of $3$-element subsets of $M_n$ is said to be fine if for any colouring of elements of $M_n$ in two colours there is a subset of the collection all three elements of which are of the same colour. For each $n \geq 5$ find the minimal possible number of the $3$-element subsets of a fine collection

A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.

Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$, $AD = 1000$, $BD = 2000$, $BC = 2001$, and $DC = 1999$. Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$. Find $AE$.

In $\vartriangle ABC$, $AB = 13$, $BC = 14,$ and $C A = 15$. Let $E$ and $F$ be the feet of the altitudes from $B$ onto $C A$, and $C$ onto $AB$, respectively. A line $\ell$ is parallel to $EF$ and tangent to the circumcircle of $ABC$ on minor arc $BC$. Let $X$ and $Y$ be the intersections of $\ell$ with $AB$ and $AC$ respectively. Find $X Y$ .

Let $R$ be the rectangle in the coordinate plane with corners $(0, 0)$, $(20, 0)$, $(20, 22)$, and $(0, 22)$, and partition $R$ into a $20\times 22$ grid of unit squares. For a given line in the coordinate plane, let its [i]pixelation[/i] be the set of grid squares in $R$ that contain part of the line in their interior. If $P$ is a point chosen uniformly at random in $R$, then compute the expected number of sets of grid squares that are pixelations of some line through $P$.

Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.

Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .

The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6$. The number of possible sets of $6$ cards that can be drawn from the deck is $6$ times the number of possible sets of $3$ cards that can be drawn. Find $n$.

Let $ABCD$ be a convex quadrilateral such that $\angle{BAD} = 90^{\circ}$ and its diagonals $AC$ and $BD$ are perpendicular. Let $M$ be the midpoint of side $CD$, and $E$ be the intersection of $BM$ and $AC$. Let $F$ be a point on side $AD$ such that $BM$ and $EF$ are perpendicular. If $CE = AF\sqrt{2}$ and $FD = CE\sqrt{2}$, show that $ABCD$ is a square.

The evil Dr. Doom seeks to destroy his enemy, the Intergalactic Federation, and has devised a plan to despin the Federation's space station. The hoop-shaped space station of mass $M$ and radius $R$ rotates once every $T$ hours to maintain artificial gravity equal to that on IPhOO. Dr. Doom plans to mount two thruster rockets, one rocket on opposite sides of the space station, to stop its rotation. Dr. Doom must accomplish his crime within a time $t$ to avoid getting caught. How much force should each rocket deliver in order to despin the Federation's space station in $t$? Express your answer in terms of $M$, $R$, $T$, $t$, and/or constants, as necessary. [i]Problem proposed by Kimberly Geddes[/i]

Find the greatest possible integer value of the side length of an equilateral triangle whose vertices belong to the interior region of a square with side length $100$.

The [i]subnumbers[/i] of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime. [i]Proposed by Lewis Chen[/i]

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$, with $AB=6$, $BC=7$, $CD=8$. Find $AD$.