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.

%%% [i]This problem intentionally left blank.[/i] %%%

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?

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

Let $O_1, O_2, ..., O_{2012}$ be $2012$ circles in the plane such that no circle intersects or contains anyother circle and no two circles have the same radius. For each $1\le i < j \le 2012$, let $P_{i,j}$ denotethe point of intersection of the two external tangent lines to $O_i$ and $O_j$, and let $T$ be the set of all $P_{i,j}$ (so $|T|=\binom {2012}{2}= 2023066$). Suppose there exists a subset $S\subset T$ with $|S|= 2021056$ such that all points in $S$ lie on the same line. Prove that all points in $T$ lie on the same line.

Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.

Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$? $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points. It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points. Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.

For every natural number $a$, consider the set $S(a)=\{a^n+a+1|n=2,3,\ldots\}$. Does there exist an infinite set $A\subset\mathbb N$ with the property that for any two distinct elements $x,y\in A$, $x$ and $y$ are coprime and $S(x)\cap S(y)=\emptyset$?

Let $x$ and $y$ be positive real numbers such that $y^3 + y \leq x - x^3$. Prove that (A) $y < x < 1$ (B) $x^2 + y^2 < 1$.

Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac mn$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$. [i]Proposed by David Altizio[/i]

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]

For any natural number $ n\ge 2, $ define $ m(n) $ to be the minimum number of elements of a set $ S $ that simultaneously satisfy: $ \text{(i)}\quad \{ 1,n\} \subset S\subset \{ 1,2,\ldots ,n\} $ $ \text{(ii)}\quad $ any element of $ S, $ distinct from $ 1, $ is equal to the sum of two (not necessarily distinct) elements from $ S. $ [b]a)[/b] Prove that $ m(n)\ge 1+\left\lfloor \log_2 n \right\rfloor ,\quad\forall n\in\mathbb{N}_{\ge 2} . $ [b]b)[/b] Prove that there are infinitely many natural numbers $ n\ge 2 $ such that $ m(n)=m(n+1). $ $ \lfloor\rfloor $ denotes the usual integer part.

Let $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ be a function such that $\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*.$ $a)$ Prove that $f(1)=1.$ $b)$ Find function $f.$

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.