For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.

Let $A = \left\{1,2,3,\cdots 72 \right\}$.Prove that you can choose $36$ element from $A$ such that the sum of those $36$ elements is equal with the sum of other $36$

Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove: [i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. [i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

Let $S$ be a set of integers such that if $a$ and $b$ are in $S$ then $3a-2b$ is also in $S$. How many ways are there to construct $S$ such that $S$ contains exactly $4$ elements in the interval $[0,40]$? [i]Proposed by Harry Kim[/i]

Let $M$ be the midpoint of $AC$ in an acute triangle $ABC$. Let $K$ be a point on the minor arc $AC$, such that $\angle AKM=90^{o}$. Let $BK \cap AM=X$ and the $A$-altitude meets $BM$ at $Y$. Show that $XY \parallel AB$.

The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

Find the positive integer $n$ that satisfies the equation $$n^2 - \lfloor \sqrt{n} \rfloor = 2018$$

Let $1 \leq k \leq n.$ Consider all finite sequences of positive integers with sum $n.$ Find $T(n,k),$ the total number of terms of size $k$ in all of the sequences.

Let $ \left( a_n \right) ,\left( b_n \right) $ be two sequences of real numbers from the interval $ (-1,1) $ having the property that $$ \max\left( \left| a_{n+1} -a_n \right| ,\left| b_{n+1} -b_n \right| \right) \le\frac{1}{(n+4)(n+5)} , $$ for any natural number. Prove that $ \left| a_nb_n -a_1b_1 \right|\le 1/2, $ for any natural number $ n. $ [i]Cristinel Mortici[/i]

$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$ $b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.

Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?

There are $2008$ blue, $2009$ red and $2010$ yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.

Real numbers $x,y$ satisfy that $(x+5)^2+(y-12)^2=14^2$, then the minumum value of $x^2+y^2$ is $\text{(A)}2\qquad\text{(B)}1\qquad\text{(C)}\sqrt3\qquad\text{(D)}\sqrt2\qquad$

All sides of a rectangle are odd positive integers. Prove that there does not exist a point inside the rectangle whose distance to each of the vertices is an integer.

Positive integer $n$ is not larger than $2000$, and $n$ is equal to the sum of no less than sixty adjacent positive integers. Then number of such numbers is________.

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors. Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations. [i]Proposed by Anthony Wang[/i]

We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers.

If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.

Define a function $f(n)$ from the positive integers to the positive integers such that $f(f(n))$ is the number of positive integer divisors of $n$. Prove that if $p$ is a prime, then $f(p)$ is prime.

1. There are $100$ participants. Out of every group of $12$ participants, there is one pair of familiar participants. Each participant is given a number (not necessarily $1$ through $100$). Prove that there is a pair of familiar participants whose number has the same starting digit. 2. $\sqrt{x + \sqrt{x + \sqrt{x + \dots + \sqrt{x}}}} = y$. If the left side is finite, find all integer solutions. 3. Is there a geometric sequence such that $a_0 > 0, b > 1$, and so that $a_l$ is an integer for $0 \le l \le 9$, but $a_l$ is not an integer for $l>9$? If so, find it. 4. Suppose r and s are positive integers and that $2^r$ is a permutation of the decimal representation of $2^s$. Prove that $r=s$. 5. Find the minimum area of a right triangle with an inscribed circle that has a radius of $1$ cm. [hide = Note]This isn't exactly verbatim, just paraphrased. I will update the questions when the official problems/solutions are released. In the meanwhile, feel free to post your solutions below![/hide]

Let there be given two lines $D_1$ and $D_2$ which intersect at point $O$, and a point $M$ not on any of these lines. Consider two variable points $A\in D_1$ and $b\in D_2$ such that $M$ belongs to the segment $AB$. (a) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Construct such points $A$ and $B$. (b) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Show that for such $A$ and $B$, the perimeters of $\triangle OAM$ and $\triangle OBM$ are equal, and that $\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}$. Construct such points $A$ and $B$.