Find all pairs of positive integers $a,b$ with $$a^a+b^b \mid (ab)^{|a-b|}-1.$$

Alicia earns $ \$20$ per hour, of which $ 1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? $ \textbf{(A)}\ 0.0029 \qquad \textbf{(B)}\ 0.029 \qquad \textbf{(C)}\ 0.29 \qquad \textbf{(D)}\ 2.9 \qquad \textbf{(E)}\ 29$

a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

Let $a_1,a_2 ,\ldots, a_n$ be an arithmetic progression of integers such that $i|a_i$ for $i=1, 2,\ldots ,n-1$ and $n\nmid a_n$. Prove that $n$ is a prime power.

Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.

In a convex quadrilateral $ABCD$ we have $\angle ADB=\angle ACD$ and $AC=CD=DB$. If the diagonals $AC$ and $BD$ intersect at $X$, prove that $\frac{CX}{BX}-\frac{AX}{DX}=1$.

The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.

In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.

Find positive reals $a,b,c$ such that: $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} = 2$$

The natural number $n>1$ is such that there exist $a\in \mathbb{N}$ and a prime number $q$ which satisfy the following conditions: 1) $q$ divides $n-1$ and $q>\sqrt{n}-1$ 2) $n$ divides $a^{n-1}-1$ 3) $gcd(a^\frac{n-1}{q}-1,n)=1$. Is it possible for $n$ to be a composite number?

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) $\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

Let $ABCD$ be a convex quadrilateral and let $O$ be the intersection of its diagonals. Let $P,Q,R,$ and $S$ be the projections of $O$ on $AB,BC,CD,$ and $DA$ respectively. Prove that \[2(OP+OQ+OR+OS)\leq AB+BC+CD+DA.\]

If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, then the sum of first 110 terms is: $\text{(A)} \ 90 \qquad \text{(B)} \ -90 \qquad \text{(C)} \ 110 \qquad \text{(D)} \ -110 \qquad \text{(E)} \ -100$