From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?

Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

Given triangle $A_0B_0C_0$. On the segment $A_0B_0$ points $A_1$, $A_2$, $...$, $A_n$, and on the segment $B_0C_0$ - points $C_1$, $C_2$, $...$, $Cn$ so that all segments $A_iC_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are parallel to each other and all segments $ C_iA_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are too. Segments $C_0A_1$, $A_1C_2$, $A_2C_1$ and $C_1A_0$ bound a certain parallelogram, segments $C_1A_2$, $A_2C_3$, $A_3C_2$ and $C_2A_1$ too, etc. Prove that the sum of the areas of all $n -1$ resulting parallelograms less than half the area of triangle $A_0B_0C_0$.

Find the area of the smallest possible square that contains the points $(2,-1)$ and $(4,4)$.

Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy \[f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} \]

Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\). [i]Proposed by Elliott Liu[/i]

A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work. Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved). a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong. b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.

Given are positive integers $r$ and $k$ and an infinite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.

Compute the sum of all positive integers $n$ such that $n^2-3000$ is a perfect square.

Find all integers $n$ such that $n^{n-1}-1$ is square-free.

A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

If $xy:yz:zx=6:8:12,$ and $x^3+y^3+z^3:xyz$ is $m:n$ where $m$ and $n$ are relatively prime positive integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

The numbers from $1$ to $500$ are written on the board. Two players $A$ and $B$ erase alternately one number at a time, and $A$ deletes the first number. If the sum of the last two number on the board is divisible by $3$, $B$ wins, otherwise $A$ wins. Which player can lay out a strategy that ensures this player's victory?

Let $S = \{f(a, b) | a, b = 1,2,3, 4$ and $a \ne b\}$, and consider all nonzero polynomials $p(X,Y )$ with integer coefficients such that $p(a, b) = 0$ for every element $(a,b)$ in $S$. (a) What is the minimal degree of such polynomial $p(X, Y )$ ? (b) Determine all such polynomials $p(X, Y )$ with minimal degree.

Prove that the side $AB$ of a convex quadrilateral $ABCD$ is less than its diagonal $AC$ if $|AB|+|BC| \le |AC| +| CD|$.

Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.

Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.

Among citizens of Cambridge there exist $8$ different types of blood antigens. In a crowded lecture hall are $256$ students, each of whom has a blood type corresponding to a distinct subset of the antigens; the remaining of the antigens are foreign to them. Quito the Mosquito flies around the lecture hall, picks a subset of the students uniformly at random, and bites the chosen students in a random order. After biting a student, Quito stores a bit of any antigens that student had. A student bitten while Quito had $k$ blood antigen foreign to him/her will suffer for $k$ hours. What is the expected total suffering of all $256$ students, in hours?

$AB, AC$ are two chords of the circle centered at $O$. The diameter, which is perpendicular to $BC$, intersects $AB, AC$ at $F, G$ respectively ($F$ is in the circle). The tangent from $G$ tangents the circle at $T$. Prove that $F$ is the projection of $T$ on $OG. $

The area of a pizza with radius $4$ inches is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? $\textbf{(A) } 25 \qquad\textbf{(B) } 33 \qquad\textbf{(C) } 44\qquad\textbf{(D) } 66 \qquad\textbf{(E) } 78$

Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting \begin{align*} p_{n+1}(x)=(p_n(x))^2-\alpha \end{align*} How many distinct real roots does $p_{10}(x)$ have?

How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?