Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is [I]small[/I] if $A, B, C$ are single-digit positive integers. It is [I]full[/I] if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

Peter Rabbit is hopping along the number line, always jumping in the positive $x$ direction. For his first jump, he starts at $0$ and jumps $1$ unit to get to the number $1.$ For his second jump, he jumps $4$ units to get to the number $5.$ He continues jumping by jumping $1$ unit whenever he is on a multiple of $3$ and by jumping $4$ units whenever he is on a number that is not a multiple of $3.$ What number does he land on at the end of his $100$th jump? $$ \mathrm a. ~ 297\qquad \mathrm b.~298\qquad \mathrm c. ~299 \qquad \mathrm d. ~300 \qquad \mathrm e. ~301 $$

Give a prime number $p>2023$. Let $r(x)$ be the remainder of $x$ modulo $p$. Let $p_1<p_2< \ldots <p_m$ be all prime numbers less that $\sqrt[4]{\frac{1}{2}p}$. Let $q_1, q_2, \ldots, q_n$ be the inverses modulo $p$ of $p_1, p_2, \ldots p_n$. Prove that for every integers $0 < a,b < p$, the sets $$\{r(q_1), r(q_2), \ldots, r(q_m)\}, ~~ \{r(aq_1+b), r(aq_2+b), \ldots, r(aq_m+b)\}$$ have at most $3$ common elements.

For every natural number $ n $, define $ s(n) $ as the smallest natural number so that for every natural number $ a $ relatively prime to $n$, this equation holds: \[ a^{s(n)} \equiv 1 (mod n) \] Find all natural numbers $ n $ such that $ s(n) = 2010 $

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.

Find all nonnegative integers $n$ for which $n^8-n^2$ is not divisible by $72$.

Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$ Prove that $x=1$. [i](Silouanos Brazitikos, Greece)[/i]

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.

Find all primes $p$ such that $p^2-p+1$ is a perfect cube.

El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?

Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.

[u]Bases[/u] Many of you may be familiar with the decimal (or base $10$) system. For example, when we say $2013_{10}$, we really mean $2\cdot 10^3+0\cdot 10^2+1\cdot 10^1+3\cdot 10^0$. Similarly, there is the binary (base $2$) system. For example, $11111011101_2 = 1 \cdot 2^{10}+1 \cdot 2^9+1 \cdot 2^8+1 \cdot 2^7+1 \cdot 2^6+0 \cdot 2^5+1 \cdot 2^4+1 \cdot 2^3+1 \cdot 2^2+0 \cdot 2^1+1 \cdot 2^0 = 2013_{10}.$ In general, if we are given a string $(a_na_{n-1} ... a_0)_b$ in base $b$ (the subscript $b$ means that we are in base $b$), then it is equal to $\sum^n_{i=0} a_ib^i$. It turns out that for every positive integer $b > 1$, every positive integer $k$ has a unique base $b$ representation. That is, for every positive integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < b$ such that $(a_na_{n-1} ... a_0)_b = k$. We can adapt this to bases $b < -1$. It actually turns out that if $b < -1$, every nonzero integer has a unique base b representation. That is, for every nonzero integer $k$, there exists a unique $n$ and digits $0 \le a_0,..., a_n < |b|$ such that $(a_na_{n-1} ... a_0)_b = k$. The next five problems involve base $-4$. Note: Unless otherwise stated, express your answers in base $10$. [b]p6.[/b] Evaluate $1201201_{-4}$. [b]p7.[/b] Express $-2013$ in base $-4$. [b]p8.[/b] Let $b(n)$ be the number of digits in the base $-4$ representation of $n$. Evaluate $\sum^{2013}_{i=1} b(i)$. [b]p9.[/b] Let $N$ be the largest positive integer that can be expressed as a $2013$-digit base $-4$ number. What is the remainder when $N$ is divided by $210$? [b]p10.[/b] Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \ne 4$ such that the base $-4$ representation of $n$ is the same as the base $b$ representation of $n$.

For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?

Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$. (Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)

It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$. [i]Proposed by C. A. Grimm [/i]

For an $n$-tuple of integers, define a transformation to be: $$(a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1)$$ Find all ordered pairs of integers $(n,k)$ with $n,k\geq 2$, such that for any $n$-tuple of integers $(a_1,a_2,\cdots,a_{n-1},a_n)$, after a finite number of transformations, every element in the of the $n$-tuple is a multiple of $k$.

Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. [i]In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png[/img]

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.