Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]

Snow White and seven dwarfs live in their house in the forest. During several days some dwarfs worked in the diamond mine, while others were collecting mushrooms. Each dwarf each day was doing only one type of job. It is known that in any two consecutive days there are exactly three dwarfs which did both types of job. Also, for any two days at least one dwarf did both types of job. What is maximum amount of days which this situation could last? [i]M. Karpuk[/i]

A dress originally priced at 80 dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was $ \text{(A)}\ \text{45 dollars}\qquad\text{(B)}\ \text{52 dollars}\qquad\text{(C)}\ \text{54 dollars}\qquad\text{(D)}\ \text{66 dollars}\qquad\text{(E)}\ \text{68 dollars} $

We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number. (Folklore)

How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities? \begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [asy] draw(circle((0,0),13)); draw(circle((5,-6.2),5)); draw(circle((-5,-6.2),5)); label("$B$", (9.5,-9.5), S); label("$A$", (-9.5,-9.5), S); [/asy] $\textbf{(A) } 21 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 69 \qquad \textbf{(E) } 93 $

There is exactly one astronomer on every planet of a certain system. He watches the closest planet. The number of the planets is odd and all of the distances are different. Prove that there one planet being not watched.

Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Suppose that there are circles $\omega_2,...,\omega_7$ such that: (a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$, (b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$. Prove that $\omega_1 = \omega_7$.

Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$. Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

[b]p1.[/b] Three positive integers sum to $16$. What is the least possible value of the sum of their squares? [b]p2.[/b] Ben is thinking of an odd positive integer less than $1000$. Ben subtracts $ 1$ from his number and divides by $2$, resulting in another number. If his number is still odd, Ben repeats this procedure until he gets an even number. Given that the number he ends on is $2$, how many possible values are there for Ben’s original number? [b]p3.[/b] Triangle $ABC$ is isosceles, with $AB = BC = 18$ and has circumcircle $\omega$. Tangents to $\omega$ at $ A$ and $ B$ intersect at point $D$. If $AD = 27$, what is the length of $AC$? [b]p4.[/b] How many non-decreasing sequences of five natural numbers have first term $ 1$, last term $ 11$, and have no three terms equal? [b]p5.[/b] Adam is bored, and has written the string “EMCC” on a piece of paper. For fun, he decides to erase every letter “C”, and replace it with another instance of “EMCC”. For example, after one step, he will have the string “EMEMCCEMCC”. How long will his string be after $8$ of these steps? [b]p6.[/b] Eric has two coins, which land heads $40\%$ and $60\%$ of the time respectively. He chooses a coin randomly and flips it four times. Given that the first three flips contained two heads and one tail, what is the probability that the last flip was heads? [b]p7.[/b] In a five person rock-paper-scissors tournament, each player plays against every other player exactly once, with each game continuing until one player wins. After each game, the winner gets $ 1$ point, while the loser gets no points. Given that each player has a $50\%$ chance of defeating any other player, what is the probability that no two players end up with the same amount of points? [b]p8.[/b] Let $\vartriangle ABC$ have $\angle A = \angle B = 75^o$. Points $D, E$, and $F$ are on sides $BC$, $CA$, and $AB$, respectively, so that $EF$ is parallel to $BC$, $EF \perp DE$, and $DE = EF$. Find the ratio of $\vartriangle DEF$’s area to $\vartriangle ABC$’s area. [b]p9.[/b] Suppose $a, b, c$ are positive integers such that $a+b =\sqrt{c^2 + 336}$ and $a-b =\sqrt{c^2 - 336}$. Find $a+b+c$. [b]p10.[/b] How many times on a $12$-hour analog clock are there, such that when the minute and hour hands are swapped, the result is still a valid time? (Note that the minute and hour hands move continuously, and don’t always necessarily point to exact minute/hour marks.) [b]p11.[/b] Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$. [b]p12.[/b] Find the value of $\sqrt{3 \sqrt{3^3 \sqrt{3^5 \sqrt{...}}}}$, where the exponents are the odd natural numbers, in increasing order. [b]p13.[/b] Jamesu and Fhomas challenge each other to a game of Square Dance, played on a $9 \times 9$ square grid. On Jamesu’s turn, he colors in a $2\times 2$ square of uncolored cells pink. On Fhomas’s turn, he colors in a $1 \times 1$ square of uncolored cells purple. Once Jamesu can no longer make a move, Fhomas gets to color in the rest of the cells purple. If Jamesu goes first, what the maximum number of cells that Fhomas can color purple, assuming both players play optimally in trying to maximize the number of squares of their color? [b]p14.[/b] Triangle $ABC$ is inscribed in circle $\omega$. The tangents to $\omega$ from $B$ and $C$ meet at $D$, and segments $AD$ and $BC$ intersect at $E$. If $\angle BAC = 60^o$ and the area of $\vartriangle BDE$ is twice the area of $\vartriangle CDE$, what is $\frac{AB}{AC}$ ? [b]p15.[/b] Fhomas and Jamesu are now having a number duel. First, Fhomas chooses a natural number $n$. Then, starting with Jamesu, each of them take turns making the following moves: if $n$ is composite, the player can pick any prime divisor $p$ of $n$, and replace $n$ by $n - p$, if $n$ is prime, the player can replace n by $n - 1$. The player who is faced with $ 1$, and hence unable to make a move, loses. How many different numbers $2 \le n \le 2019$ can Fhomas choose such that he has a winning strategy, assuming Jamesu plays optimally? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

It is known about the triangle $ABC$ that $3 BC = CA + AB$. Let the $A$-symmedian of triangle $ABC$ intersect the circumcircle of triangle $ABC$ at point $D$. Prove that $\frac{1}{BD}+ \frac{1}{CD}= \frac{6}{AD}$. (Ercole Suppa, Italy)

Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$

The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.

Kelvin the Nanofrog is visiting his friend, Alex the Nanokat, who lives 483 nanometers away. On his trip to Alex's home, Kelvin travels at $k$ nanometers an hour, where $k$ is an integer, and completes the trip in an integer number of minutes. On his return journey, he travels slower by $7$ nanometers an hour, and completes the trip in an integer number of minutes. What is the smallest total number of minutes Kelvin could have spent traveling?

Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$. Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$. Show that $T_p$ has finitely many elements.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

If it is known that the equation $$x^4+ax^3+2x^2+bx+1=0$$ has a (real) root, prove the inequality $$a^2+b^2 \ge 8.$$ (A Egorov)

Prove that the following equation has no solution in integer numbers: \[ x^2 + 4 = y^5. \] [i]Bulgaria[/i]

Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$? (Boris Frenkin)