For any interval $\mathcal{A}$ in the real number line not containing zero, define its [i]reciprocal[/i] to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[m,n]$ is $10^{10}$ times the length of its reciprocal. [i]Proposed by David Altizio[/i]

Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino? [asy] size(4cm); draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0)); [/asy]

Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

(a) Let $k,n\ge 1$.Find the number of sequences $\phi=S_0,S_1,\ldots,S_k$ of subsets of $[n]=\{1,2,3,\ldots,n\}$ if for all $1\le i\le k$ we have either (i)$S_{i-1}\subset S_i$ and $|S_i-S_{i-1}|$,or (ii)$S_i\subset S_{i-1}$ and $|S_{i-1}-S_i|=1$. (b) Suppose that we add the additional condition that $S_k=\phi$.Show that now the number $f_k(n)$ of sequences is given by$f_k(n)=\frac{1}{2^n}\sum_{i=0}^n\binom ni (n-2i)^k$. Note that $f_k(n)=0$ if $k$ is odd.

Determine all graphs $G$ with the following two properties: $\bullet$ G contains at least one Hamilton path. $\bullet$ For any pair of vertices, $u, v \in G$, if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$

Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$. Let $N$ be the reflection of $M$ in $BC$. The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$. Determine the value of $\angle AKC$. (A.Blinkov)

If $ a,b,n$ are positive integers, number of solutions of the equaition $ a^2 \plus{} b^4 \equal{} 5^n$ is $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$. 1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$? 2) What is the number of students from club $A$?

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Real numbers $ a$ and $ b$ satisfy the equations $ 3^{a} \equal{} 81^{b \plus{} 2}$ and $ 125^{b} \equal{} 5^{a \minus{} 3}$. What is $ ab$? $ \textbf{(A)} \minus{} \!60 \qquad \textbf{(B)} \minus{} \!17 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 60$

Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$

Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.

Is it true that for any polynomial $P(x)$ with real coefficients of degree $2023$, there exists a natural number $n$ such that the equation $P(x) = n^{-100}$ has no rational root?

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? $\text{(A)}\ 12\% \qquad \text{(B)}\ 20\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 60\% \qquad \text{(E)}\ 66\frac{2}{3}\% $

Let $S$ be a set and let $\circ$ be a binary operation on $S$ satisfying two laws $$x\circ x=x \text{ for all } x \text{ in } S, \text{ and}$$ $$(x \circ y) \circ z= (y\circ z) \circ x \text{ for all } x,y,z \text{ in } S.$$ Show that $\circ$ is associative and commutative.

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

[b]p1.[/b] Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle. [b]p2.[/b] Determine the greatest integer that will divide $13,511$, $13,903$ and $14,589$ and leave the same remainder. [b]p3.[/b] Suppose $A, B$ and $C$ are the angles of the triangle. Show that $\cos^2 A + \cos^2 B + \cos^2 C + 2 \cos A \cos B \cos C = 1$ [b]p4.[/b] Given the linear fractional transformation $f_1(x) =\frac{2x - 1}{x + 1}$. Define $f_{n+1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3,...$ . It can be shown that $f_{35} = f_5$. (a) Find a function $g$ such that $f_1(g(x)) = g(f_1(x)) = x$. (b) Find $f_{28}$. [b]p5.[/b] Suppose $a$ is a complex number such that $a^{10} + a^5 + 1 = 0$. Determine the value of $a^{2005} + \frac{1}{a^{2005}}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?