The graphs of $y=\log_3x$, $y=\log_x3$, $y=\log_{\frac13}x$, and $y=\log_x\frac13$ are plotted on the same set of axes. How many points in the plane with positive $x-$coordinates lie on two or more of the graphs? $\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6$

In a set $X$ of n people, some know each other and others do not, where the relationship to know is symmetric; that is, if $ A$ knows $ B$. then $ B$ knows $ A$. On the other hand, given any$ 4$ people: $A, B, C$ and $D$: if $A$ knows $B$, $B$ knows $C$ and $C$ knows $D$, then it happens at least one of the following three: $A$ knows $C, B$ knows $D$ or $A$ knows $D$. Prove that $X$ can be partition into two sets $Y$ and $Z$ so that all elements of $Y$ know all those of $Z$ or no element in $Y$ knows any in $Z$.

[b]a)[/b] Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers. [b]b)[/b] Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.

Consider the four lines $y = mx-k^2$ for different integer $k$. Let $(x_i,y_i)$, $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$. Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$.

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]

$S$ is a subset of Natural numbers which has infinite members. $$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$ Prove the set of prime divisors of $S’$ has also infinite members [i]Proposed by Yahya Motevassel[/i]

Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$. $ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer. $ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.

There's a group of $21$ people such that each person has no more than $7$ friends among the others and any two friends have a different number of total friends. Prove that there are $6$ people, none of which knows the others.

Find, with proof, all pairs of positive integers $(n,d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of $n$ integers $a_1,a_2,...,a_n$ such that $a_1 + a_2 + ... + a_n = S$ and $a_n-a_1=d.$

Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at his or her own roll, and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say $19$, $22$, $21$, and $23$, respectively. Compute the product of the dice rolls.

Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers. (Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)

Consider an infinite plane divided into unit squares by horizontal and vertical lines. A coloring of some cells in this grid is called a $\emph{net coloring}$ if the centers of the colored squares coincide with the intersection points of an infinite family of equally spaced parallel lines and another directed and equally spaced infinite family of lines. The distance between the centers of the nearest colored squares is called the size of the $\emph{net coloring}.$ Determine all natural numbers \(N\) for which it is possible to color all these unit squares using \(N\) colors such that the following conditions are met: $\bullet$ Each color is used to color at least one square. $\bullet$The coloring for every color forms a $\emph{net coloring}.$ $\bullet$ The sizes of each of the \(N\) $\emph{net colorings}$ are equal. [i]Proposed by Aleksandre Saatashvili, Georgia [/i]

Let $n$ be a positive integer, and let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. Alex the Kat writes down the $n^2$ numbers of the form $\min(a_i, a_j)$, and Kelvin the Frog writes down the $n^2$ numbers of the form $\max(b_i, b_j)$. Let $x_n$ be the largest possible size of the set $\{a_1, \ldots, a_n, b_1, \ldots, b_n\}$, such that Alex the Kat and Kelvin the Frog write down the same collection of numbers. Determine the number of distinct integers in the sequence $x_1, x_2, \ldots, x_{10,000}$.

In the tetrahedron $ABCD$, perpendiculars $AB'$, $AC'$, $AD'$ are dropped from vertex $A$, on the plane dividing the dihedral angles at the edges $CD$, $BD$, $BC$ in half. Prove that the plane $(B'C'D' )$ is parallel to the plane $(BCD)$.

Let $N_2$ be the answer to problem 2. On a number line, Tanya circles the first $\ell$ positive integers. Then, starting with the greatest number in the most recent circle, she circles the next $\ell$ positive integers, so that the two circles have exactly one number in common; she repeats this until $N_2$ is in a circle. Compute the sum of all possible values of $\ell$ for which $N_2$ is the greatest number in a circle.

Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ ([i]Dan Schwarz[/i])