There are coins in denominations of $a$ and $b$ doubloons, where $a$ and $b$ are given mutually prime natural numbers, with $a < b < 100$. A non-negative integer $n$ is called [i]lucky[/i] if the sum in $n$ doubloons can be scored with using no more than $1000$ coins. Find the number of lucky numbers. [i]From the folklore[/i]

[i]20 problems for 20 minutes. [/i] [b]p1.[/b] Evaluate $\frac{\sqrt2 \cdot \sqrt6}{\sqrt3}.$ [b]p2.[/b] If $6\%$ of a number is $1218$, what is $18\%$ of that number? [b]p3.[/b] What is the median of $\{42, 9, 8, 4, 5, 1,13666, 3\}$? [b]p4.[/b] Define the operation $\heartsuit$ so that $i\heartsuit u = 5i - 2u$. What is $3\heartsuit 4$? p5. How many $0.2$-inch by $1$-inch by $1$-inch gold bars can fit in a $15$-inch by $12$-inch by $9$-inch box? [b]p6.[/b] A tetrahedron is a triangular pyramid. What is the sum of the number of edges, faces, and vertices of a tetrahedron? [b]p7.[/b] Ron has three blue socks, four white socks, five green socks, and two black socks in a drawer. Ron takes socks out of his drawer blindly and at random. What is the least number of socks that Ron needs to take out to guarantee he will be able to make a pair of matching socks? [b]p8.[/b] One segment with length $6$ and some segments with lengths $10$, $8$, and $2$ form the three letters in the diagram shown below. Compute the sum of the perimeters of the three figures. [img]https://cdn.artofproblemsolving.com/attachments/1/0/9f7d6d42b1d68cd6554d7d5f8dd9f3181054fa.png[/img] [b]p9.[/b] How many integer solutions are there to the inequality $|x - 6| \le 4$? [b]p10.[/b] In a land for bad children, the flavors of ice cream are grass, dirt, earwax, hair, and dust-bunny. The cones are made out of granite, marble, or pumice, and can be topped by hot lava, chalk, or ink. How many ice cream cones can the evil confectioners in this ice-cream land make? (Every ice cream cone consists of one scoop of ice cream, one cone, and one topping.) [b]p11.[/b] Compute the sum of the prime divisors of $245 + 452 + 524$. [b]p12.[/b] In quadrilateral $SEAT$, $SE = 2$, $EA = 3$, $AT = 4$, $\angle EAT = \angle SET = 90^o$. What is the area of the quadrilateral? [b]p13.[/b] What is the angle, in degrees, formed by the hour and minute hands on a clock at $10:30$ AM? [b]p14.[/b] Three numbers are randomly chosen without replacement from the set $\{101, 102, 103,..., 200\}$. What is the probability that these three numbers are the side lengths of a triangle? [b]p15.[/b] John takes a $30$-mile bike ride over hilly terrain, where the road always either goes uphill or downhill, and is never flat. If he bikes a total of $20$ miles uphill, and he bikes at $6$ mph when he goes uphill, and $24$ mph when he goes downhill, what is his average speed, in mph, for the ride? [b]p16.[/b] How many distinct six-letter words (not necessarily in any language known to man) can be formed by rearranging the letters in $EXETER$? (You should include the word EXETER in your count.) [b]p17.[/b] A pie has been cut into eight slices of different sizes. Snow White steals a slice. Then, the seven dwarfs (Sneezy, Sleepy, Dopey, Doc, Happy, Bashful, Grumpy) take slices one by one according to the alphabetical order of their names, but each dwarf can only take a slice next to one that has already been taken. In how many ways can this pie be eaten by these eight persons? [b]p18.[/b] Assume that $n$ is a positive integer such that the remainder of $n$ is $1$ when divided by $3$, is $2$ when divided by $4$, is $3$ when divided by $5$, $...$ , and is $8$ when divided by $10$. What is the smallest possible value of $n$? [b]p19.[/b] Find the sum of all positive four-digit numbers that are perfect squares and that have remainder $1$ when divided by $100$. [b]p20.[/b] A coin of radius $1$ cm is tossed onto a plane surface that has been tiled by equilateral triangles with side length $20\sqrt3$ cm. What is the probability that the coin lands within one of the triangles? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there exist infinitely many positive integers $k$ such that the equation $$\frac{n^2+m^2}{m^4+n}=k$$ don't have any positive integer solution.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

Say a number is [i]tico[/i] if the sum of it's digits is a multiple of $2003$. $\text{(i)}$ Show that there exists a positive integer $N$ such that the first $2003$ multiples, $N,2N,3N,\ldots 2003N$ are all tico. $\text{(ii)}$ Does there exist a positive integer $N$ such that all it's multiples are tico?

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?

Prove that, for any integer $n\geq 2$, there exists an integer $x$ such that $3^n|x^3+2017$, but $3^{n+1}\not | x^3+2017$.

Determine all positive integers $n$ such that $\frac{n(n-1)}{2}-1$ divides $1^7+2^7+\dots +n^7$.

Find solution in positive integers : $$n^5+n^4=7^m-1$$

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$

1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$. 1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black. 1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$x^2-y^2+2y(f(x)+f(y))$$ is a square of an integer for all positive integers $x$ and $y$.

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$ Where $f^n(m)$ is defined in the standard iterative manner.

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.

Let $a,b,d$ be integers such that $\left|a\right| \geqslant 2$, $d \geqslant 0$ and $b \geqslant \left( \left|a\right| + 1\right)^{d + 1}$. For a real coefficient polynomial $f$ of degree $d$ and integer $n$, let $r_n$ denote the residue of $\left[ f(n) \cdot a^n \right]$ mod $b$. If $\left \{ r_n \right \}$ is eventually periodic, prove that all the coefficients of $f$ are rational.

Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$

For $a,n\in\mathbb{Z}^+$, consider the following equation: \[ a^2x+6ay+36z=n\quad (1) \] where $x,y,z\in\mathbb{N}$. a) Find all $a$ such that for all $n\geq 250$, $(1)$ always has natural roots $(x,y,z)$. b) Given that $a>1$ and $\gcd (a,6)=1$. Find the greatest value of $n$ in terms of $a$ such that $(1)$ doesn't have natural root $(x,y,z)$.

Show that there are infinitely many triples of distinct positive integers $a, b, c$ such that each divides the product of the other two and $a + b = c + 1$.