A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.

Tetris is a Soviet block game developed in $1984$, probably to torture misbehaving middle school children. Nowadays, Tetris is a game that people play for fun, and we even have a mini-event featuring it, but it shall be used on this test for its original purpose. The $7$ Tetris pieces, which will be used in various problems in this theme, are as follows: [img]https://cdn.artofproblemsolving.com/attachments/b/c/f4a5a2b90fcf87968b8f2a1a848ad32ef52010.png[/img] [b]p1.[/b] Each piece has area $4$. Find the sum of the perimeters of each of the $7$ Tetris pieces. [b]p2.[/b] In a game of Tetris, Qinghan places $4$ pieces every second during the first $2$ minutes, and $2$ pieces every second for the remainder of the game. By the end of the game, her average speed is $3.6$ pieces per second. Find the duration of the game in seconds. [b]p3.[/b] Jeff takes all $7$ different Tetris pieces and puts them next to each other to make a shape. Each piece has an area of $4$. Find the least possible perimeter of such a shape. [b]p4.[/b] Qepsi is playing Tetris, but little does she know: the latest update has added realistic physics! She places two blocks, which form the shape below. Tetrominoes $ABCD$ and $EFGHI J$ are both formed from $4$ squares of side length $1$. Given that $CE = CF$, the distance from point $I$ to the line $AD$ can be expressed as $\frac{A\sqrt{B}-C}{D}$ . Find $1000000A+10000B +100C +D$. [img]https://cdn.artofproblemsolving.com/attachments/9/a/5e96a855b9ebbfd3ea6ebee2b19d7c0a82c7c3.png[/img] [b]p5.[/b] Using the following tetrominoes: [img]https://cdn.artofproblemsolving.com/attachments/3/3/464773d41265819c4f452116c1508baa660780.png[/img] Find the number of ways to tile the shape below, with rotation allowed, but reflection disallowed: [img]https://cdn.artofproblemsolving.com/attachments/d/6/943a9161ff80ba23bb8ddb5acaf699df187e07.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a right triangle $ABC $ with a right angle $\angle C $, n the sides $AC$ and $AB$, the points $M$ and $N$ are selected, respectively, that $CM = MN$ and $\angle MNB = \angle CBM$. Let the point $K$ be the projection of the point $C $ on the segment $MB $. Prove that the line $NK$ passes through the midpoint of the segment $BC$. (Alex Klurman)

Can a $5\times 7$ checkerboard be covered by L's (figures formed from a $2\times2$ square by removing one of its four $1\times1$ corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's? [i]M. Evdokimov[/i]

Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Approximately what percent of the design is black? [asy] real d=320; pair O=origin; pair P=O+8*dir(d); pair A0 = origin; pair A1 = O+1*dir(d); pair A2 = O+2*dir(d); pair A3 = O+3*dir(d); pair A4 = O+4*dir(d); pair A5 = O+5*dir(d); filldraw(Circle(A0, 6), white, black); filldraw(circle(A1, 5), black, black); filldraw(circle(A2, 4), white, black); filldraw(circle(A3, 3), black, black); filldraw(circle(A4, 2), white, black); filldraw(circle(A5, 1), black, black); [/asy] $ \textbf{(A)}\ 42\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 48\qquad$

A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

$P$ is a point on arc $\overarc{BC}$ of the circumcircle of $\triangle ABC$ not containing $A$, $K$ lies on segment $AP$ such that $BK$ bisects $\angle ABC$. The circumcircle of $\triangle KPC$ meets $AC,BD$ at $D,E$ respectively. $PE$ meets $AB$ at $F$. Prove that $\angle ABC=2\angle FCB$.

Show that there do not exist five consecutive integers whose sum of squares is itself a perfect square.

The table below shows the distance $ s$ in feet a ball rolls down an inclined plane in $ t$ seconds. \[ \begin{tabular}{|c|c|c|c|c|c|c|} \hline t & 0 & 1 & 2 & 3 & 4 & 5\\ \hline s & 0 & 10 & 40 & 90 & 160 & 250\\ \hline \end{tabular} \] The distance $ s$ for $ t \equal{} 2.5$ is: $ \textbf{(A)}\ 45\qquad \textbf{(B)}\ 62.5\qquad \textbf{(C)}\ 70\qquad \textbf{(D)}\ 75\qquad \textbf{(E)}\ 82.5$

Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that: $$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$

For each $q\in\mathbb Q$, let $\pi(q)$ denote the period of the repeating base-$16$ expansion of $q$, with the convention of $\pi(q)=0$ if $q$ has a terminating base-$16$ expansion. Find the maximum value among \[\pi\left(\frac11\right),~\pi\left(\frac12\right),~\dots,~\pi\left(\frac1{70}\right).\]

Find all functions $ f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying \[ f(f(x) \plus{} y) \equal{} f(x^2 \minus{} y) \plus{} 4f(x)y \] for all $ x,y \in \mathbb{R}$.

The incircle of triangle $ABC$ with $AB\neq AC$ has center $I$ and is tangent to $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. The circumcircle of triangle $ADI$ intersects $AB$ and $AC$ again at $X$ and $Y.$ Prove that $EF$ bisects $XY.$

Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by \[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\] Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.

Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.

Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd. [i]A. Poliansky[/i]

A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system: $\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$. $\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don’t know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values. $\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal. They then provide you with the following equations, written in Gliesian, which are known to be true: [img]https://cdn.artofproblemsolving.com/attachments/b/e/e2e44c257830ce8eee7c05535046c17ae3b7e6.png[/img]

The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours. [i]Proposed by S. Berlov, S. Ivanov[/i]

Let $ g(n,k)$ denote the number of strongly connected, $ \textit{simple}$ directed graphs with $ n$ vertices and $ k$ edges. ($ \textit{Simple}$ means no loops or multiple edges.) Show that \[ \sum_{k=n}^{n^2-n}(-1)^kg(n,k)=(n-1)!.\] [i]A. A. Schrijver[/i]

Let $ABCD$ and $A_1B_1C_1D_1$ be convex quadrangles in a plane, such that $AB=A_1B_1$, $BC=B_1C_1$, $CD=C_1D_1$ and $DA=D_1A_1$. Given that diagonals $AC$ and $BD$ are perpendicular to each other, prove that the same holds for diagonals $A_1C_1$ and $B_1D_1$.

In a dance class there are $10$ boys and $10$ girls. It is known that for each $1\le k\le 10$ and for each group of $k$ boys, the number of girls who are friends with at least one boy in the group is not less than $k$. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.

Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.