On some cells of a $2n \times 2n$ board are placed white and black markers (at most one marker on every cell). We first remove all black markers which are in the same column with a white marker, then remove all white markers which are in the same row with a black one. Prove that either the number of remaining white markers or that of remaining black markers does not exceed $n^2$.

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.

Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. [i]Proposed by Finbarr Holland, Ireland[/i]

In the following system of equations $$|x+y|+|y|=|x-1|+|y-1|=2,$$ find the sum of all possible $x$.

$19.$ The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37 ?$

There are $n\geq2$ numbers $x_1, x_2, \ldots, x_n$ such that $x^2_i=1 (1\leq i\leq n)$ and $$x_1x_2+x_2x_3+\dots+x_{n-1}x_n+x_nx_1=0.$$ Prove that $n$ is divisible with $4$.

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be $ 6$. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts $ 1$. If it comes up tails, he takes half of the previous term and subtracts $ 1$. What is the probability that the fourth term in Jacob's sequence is an integer? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{5}{8} \qquad \textbf{(E)}\ \frac{3}{4}$

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

Let $f(x) = x^2 +6x+6.$ Compute the greatest real number $x$ such that $f(f(f(f(f(f(x)))))) = 0.$

Point $E$ is on side $CD$ of rectangle $ABCD$ such that $\frac{CE}{DE} =\frac{2}{5}.$ If the area of triangle $BCE$ is $30$, what is the area of rectangle $ABCD$?

If $a_1/b_1=a_2/b_2=a_3/b_3$ and $p_1,p_2,p_3$ are not all zero, show that for all $n\in\mathbb{N}$, \[ \left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}. \]

Let $P$ be a finite set of primes, $A$ an infinite set of positive integers, where every element of $A$ has a prime factor not in $P$. Prove that there exist an infinite subset $B$ of $A$, such that the sum of elements in any finite subset of $B$ has a prime factor not in $P$.

Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^{\circ}.$ What is the square of the distance (in feet) between Abdul and Bharat? $\textbf{(A) } 1728 \qquad\textbf{(B) } 2601 \qquad\textbf{(C) } 3072 \qquad\textbf{(D) } 4608 \qquad\textbf{(E) } 6912$

Find the largest interval $ M \subseteq \mathbb{R^ \plus{} }$, such that for all $ a$, $ b$, $ c$, $ d \in M$ the inequality \[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}\] holds. Does the inequality \[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}\] hold too for all $ a$, $ b$, $ c$, $ d \in M$? ($ \mathbb{R^ \plus{} }$ denotes the set of positive reals.)

For odd numbers $a_1 , ..., a_k$ and even numbers $b_1 , ..., b_k$ , we know that $\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n$ is satisfied for n = 1,2, ..., N. Prove that $k\geq 2^N$ and that for $k = 2^N$ there exists a solution $(a_1,...,b_1,...)$ with the above properties.

At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.

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Given a string of length $2n$, we perform the following operation: [list] [*]Place all the even indexed positions together, and then all the odd indexed positions next. Indexing is done starting from $0$.[/*] [/list] For example, say our string is ``abcdef''. Performing our operation yields ``abcdef'' $\to$ ``acebdf''. Performing the operation again yields ``acebdf'' $\to$ ``aedcbf''. Doing this repeatedly, we have:\\ ``abcdef'' $\to$ ``acebdf'' $\to$ ``aedcbf'' $\to$ ``adbecf'' $\to$ ``abcdef''.\\\\ You can assume that the characters in the string will be unique. It can be shown that, by performing the above operation a finite number of times we can get back our original string.\\\\ Given $n$, you have to determine the minimum number of times the operation must be performed to get our original string of length $2n$ back.\\\\ In the example given above, $2n = 6$. The minimum steps required is $4$.

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

There are $N{}$ points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum $N{}$ for which this is possible? [i]Proposed by E. Bakaev[/i]

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

A group of n  friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?