Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? [asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle); draw(card, linewidth(1)); draw(shift(2.5,0)*card, linewidth(1)); draw(shift(5,0)*card, linewidth(1)); label("$44$", (1,1.5)); label("$59$", shift(2.5,0)*(1,1.5)); label("$38$", shift(5,0)*(1,1.5));[/asy] $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Find all non-negative integer solutions of the equation \[2^x + 3^y = z^2 .\]

Let $k$ and $n$ be two given distinct positive integers greater than $1$. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase $k$ consecutive elements of an arithmetic sequence with a difference not divisible by $k$. Similarly, Nándor is allowed to erase $n$ consecutive elements of an arithmetic sequence with a difference that is not divisible by $n$. The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least $\varphi(n)+\varphi(k)$. [i]Proposed by Boldizsár Varga, Budapest[/i]

Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition \[ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} \] for all $a,b \in \mathbb{Z}_{p^n}.$

Find the largest positive integer $n$ that cannot be written as $n = 20a + 28b + 35c$ for nonnegative integers $a, b$, and $c$.

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

A possitive integer is called [i]special[/i] if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$-digit [i]special[/i] numbers (b) Are there $2000$-digit [i]special[/i] numbers?

We say a non-negative integer $n$ "[i]contains[/i]" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ [i]contains[/i] $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not [i]contain[/i] a multiple of $7$.

A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.

$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$. $$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.

Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations $$a^3 - 2a = abc - b - c$$ $$b^3 - 2b = bca - c - a$$ $$c^3 - 2c = cab - a - b.$$

The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions: (i) $6p$ does not divide $q-1$; (ii) $q$ divides $x^2+xy+y^2$; (iii) $q$ does not divide $x+y-z$. Prove that $x^p+y^p\ne z^p$.

Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after $n$ years. Find $n$.

[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test? [b]p2.[/b] You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either $20$ heads, or $16$ tails. What is the maximum number of times that you could have flipped the coin? [b]p3.[/b] The width of a rectangle is half of its length. Its area is $98$ square meters. What is the length of the rectangle, in meters? [b]p4.[/b] Carol is twice as old as her younger brother, and Carol's mother is $4$ times as old as Carol is. The total age of all three of them is $55$. How old is Carol's mother? [b]p5.[/b] What is the sum of all two-digit multiples of $9$? [b]p6.[/b] The number $2016$ is divisible by its last two digits, meaning that $2016$ is divisible by $16$. What is the smallest integer larger than $2016$ that is also divisible by its last two digits? [b]p7.[/b] Let $Q$ and $R$ both be squares whose perimeters add to $80$. The area of $Q$ to the area of $R$ is in a ratio of $16 : 1$. Find the side length of $Q$. [b]p8.[/b] How many $8$-digit positive integers have the property that the digits are strictly increasing from left to right? For instance, $12356789$ is an example of such a number, while $12337889$ is not. [b]p9.[/b] During a game, Steve Korry attempts $20$ free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with $84\%$ accuracy, assuming he makes them all? [b]p10.[/b] How many dierent ways are there to arrange the letters $MILKTEA$ such that $TEA$ is a contiguous substring? For reference, the term "contiguous substring" means that the letters $TEA$ appear in that order, all next to one another. For example, $MITEALK$ would be such a string, while $TMIELKA$ would not be. [b]p11.[/b] Suppose you roll two fair $20$-sided dice. What is the probability that their sum is divisible by $10$? [b]p12.[/b] Suppose that two of the three sides of an acute triangle have lengths $20$ and $16$, respectively. How many possible integer values are there for the length of the third side? [b]p13.[/b] Suppose that between Beijing and Shanghai, an airplane travels $500$ miles per hour, while a train travels at $300$ miles per hour. You must leave for the airport $2$ hours before your flight, and must leave for the train station $30$ minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities? [b]p14.[/b] How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of $16$? Two triangles are considered distinct if they are not congruent. [b]p15.[/b] John can drive $100$ miles per hour on a paved road and $30$ miles per hour on a gravel road. If it takes John $100$ minutes to drive a road that is $100$ miles long, what fraction of the time does John spend on the paved road? [b]p16.[/b] Alice rolls one pair of $6$-sided dice, and Bob rolls another pair of $6$-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice? [b]p17.[/b] When $20^{16}$ is divided by $16^{20}$ and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included. [b]p18.[/b] Suppose you have a $20 \times 16$ bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than $5$. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a $2\times 2$ bar and wish to break it entirely into $1\times 1$ bars. We can break it once to get two $2\times 1$ bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size $1\times 1$, and we end up using $3$ breaks in total. [b]p19.[/b] A class of $10$ students decides to form two distinguishable committees, each with $3$ students. In how many ways can they do this, if the two committees can have no more than one student in common? [b]p20.[/b] You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from $0$ to $10$ inclusive, and that no pair of vertices can share the same $x$ or $y$ coordinate value (so for example, you could not use both $(1, 2)$ and $(1, 4)$ in your polygon, but $(1, 2)$ and $(2, 1)$ is fine). What is the largest possible area that your polygon can have? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number. Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

[u]Round 5[/u] [b]p13.[/b] Initially, the three numbers $20$, $201$, and $2016$ are written on a blackboard. Each minute, Zhuo selects two of the numbers on the board and adds $1$ to each. Find the minimum $n$ for which Zhuo can make all three numbers equal to $n$. [b]p14.[/b] Call a three-letter string rearrangeable if, when the first letter is moved to the end, the resulting string comes later alphabetically than the original string. For example, $AAA$ and $BAA$ are not rearrangeable, while $ABB$ is rearrangeable. How many three-letters strings with (not necessarily distinct) uppercase letters are rearrangeable? [b]p15.[/b] Triangle $ABC$ is an isosceles right triangle with $\angle C = 90^o$ and $AC = 1$. Points $D$, $E$ and $F$ are chosen on sides $BC$,$CA$ and $AB$, respectively, such that $AEF$, $BFD$, $CDE$, and $DEF$ are isosceles right triangles. Find the sum of all distinct possible lengths of segment $DE$. [u]Round 6[/u] [b]p16.[/b] Let $p, q$, and $r$ be prime numbers such that $pqr = 17(p + q + r)$. Find the value of the product $pqr$. [b]p17.[/b] A cylindrical cup containing some water is tilted $45$ degrees from the vertical. The point on the surface of the water closest to the bottom of the cup is $6$ units away. The point on the surface of the water farthest from the bottom of the cup is $10$ units away. Compute the volume of the water in the cup. [b]p18.[/b] Each dot in an equilateral triangular grid with $63$ rows and $2016 = \frac12 \cdot 63 \cdot 64$ dots is colored black or white. Every unit equilateral triangle with three dots has the property that exactly one of its vertices is colored black. Find all possible values of the number of black dots in the grid. [u]Round 7[/u] [b]p19.[/b] Tomasz starts with the number $2$. Each minute, he either adds $2$ to his number, subtracts $2$ from his number, multiplies his number by $2$, or divides his number by $2$. Find the minimum number of minutes he will need in order to make his number equal $2016$. [b]p20.[/b] The edges of a regular octahedron $ABCDEF$ are painted with $3$ distinct colors such that no two edges with the same color lie on the same face. In how many ways can the octahedron be painted? Colorings are considered different under rotation or reflection. [b]p21.[/b] Jacob is trapped inside an equilateral triangle $ABC$ and must visit each edge of triangle $ABC$ at least once. (Visiting an edge means reaching a point on the edge.) His distances to sides $AB$, $BC$, and $CA$ are currently $3$, $4$, and $5$, respectively. If he does not need to return to his starting point, compute the least possible distance that Jacob must travel. [u]Round 8[/u] [b]p22.[/b] Four integers $a, b, c$, and $d$ with a $\le b \le c \le d$ satisfy the property that the product of any two of them is equal to the sum of the other two. Given that the four numbers are not all equal, determine the $4$-tuple $(a, b, c, d)$. [b]p23.[/b] In equilateral triangle $ABC$, points $D$,$E$, and $F$ lie on sides $BC$,$CA$ and $AB$, respectively, such that $BD = 4$ and $CD = 5$. If $DEF$ is an isosceles right triangle with right angle at $D$, compute $EA + FA$. [b]p24.[/b] On each edge of a regular tetrahedron, four points that separate the edge into five equal segments are marked. There are sixteen planes that are parallel to a face of the tetrahedron and pass through exactly three of the marked points. When the tetrahedron is cut along each of these sixteen planes, how many new tetrahedrons are produced? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934049p26256220]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5[/u] [b]p13.[/b] Mandy is baking cookies. Her recipe calls for $N$ grams of flour, where $N$ is the number of perfect square divisors of $20! + 24!$. Find $N$. [b]p14.[/b] Consider a circular table with center $R$. Beef-loving Bryan places a steak at point $I$ on the circumference of the table. Then he places a bowl of rice at points $C$ and $E$ on the circumference of the table such that $CE \parallel IR$ and $\angle ICE = 25^o$. Find $\angle CIE$. [b]p15.[/b] Enya writes the $4$-letter words $LEEK$, $BEAN$, $SOUP$, $PEAS$, $HAMS$, and $TACO$ on the board. She then thinks of one of these words and gives Daria, Ava, Harini, and Tiffany a slip of paper containing exactly one letter from that word such that if they ordered the letters on their slips correctly, they would form the word. Each person announces at the same time whether they know the word or not. Ava, Harini, and Tiffany all say they do not know the word, while Daria says she knows the word. After hearing this, Ava, Harini, and Tiffany all know the word. Assuming all four girls are perfect logicians and they all thought of the same correct word, determine Daria’s letter. [u]Round 6[/u] [b]p16.[/b] Michael receives a cheese cube and a chocolate octahedron for his 5th birthday. On every day after, he slices off each corner of his cheese and chocolate with a knife. Each slice cuts off exactly one corner. He then eats each corner sliced off. Find the difference between the total number of cheese and chocolate pieces he has eaten by the end of his $6$th birthday. (Michael’s $5$th and $6$th birthdays do not occur on leap years.) [b]p17.[/b] Let $D$ be the average of all positive integers n satisfying $$lcm (gcd (n, 2000), gcd (n, 24)) = gcd (lcm (n, 2000), lcm (n, 24)).$$ Find $3D$. [b]p18.[/b] The base $\vartriangle ABC$ of the triangular pyramid $PABC$ is an equilateral triangle with a side length of $3$. Given that $PA = 3$, $PB = 4$, and $PC = 5$, find the circumradius of $PABC$. [u]Round 7[/u] [b]p19.[/b] $2049300$ points are arranged in an equilateral triangle point grid, a smaller version of which is shown below, such that the sides contain $2024$ points each. Peter starts at the topmost point of the grid. At $9:00$ am each day, he moves to an adjacent point in the row below him. Derrick wants to prevent Peter from reaching the bottom row, so at $12:00$ pm each day, he selects a point on the bottom row and places a rock at that point. Peter stops moving as soon as he is guaranteed to end up at a point with a rock on it. At least how many moves will Peter complete, no matter how Derrick places the rocks? [img]https://cdn.artofproblemsolving.com/attachments/f/a/346d25a5d7bb7a5fbefae7edad727965312b25.png[/img] [b]p20.[/b] There are $N$ stones in a pile, where $N$ is a positive integer. Ava and Anika take turns playing a game, with Ava moving first. If there are n stones in the pile, a move consists of removing $x$ stones, where $1 < gcd(x, n) \le x < n$. Whoever first has no possible moves on their turn wins. Both Ava and Anika play optimally. Find the $2024$th smallest value of $N$ for which Ava wins. [b]p21.[/b] Alan is bored and alone, so he plays a fun game with himself. He writes down all quadratic polynomials with leading coefficient $1$ whose coefficients are integers between $-10$ and $10$, inclusive, on a blackboard. He then erases all polynomials which have a non-integer root. Alan defines the size of a polynomial $P(x)$ to be $P(1)$ and spends an hour adding up the sizes of all the polynomials remaining on the blackboard. Assuming Alan does computation perfectly, find the sum Alan obtains. [u]Round 8[/u] [b]p22.[/b] A prime number is a positive integer with exactly two distinct divisors. You must submit a prime number for this problem. If you do not submit a prime number, you gain $0$ points, and your submission will not be considered valid. The median of all valid submitted numbers is $M$ (duplicates are counted). Estimate $2M$. If your team’s absolute difference between $2M$ and your submission is the $i$th smallest absolute difference among all teams, you gain max$(23 - 2i, 0)$ points. All teams who did not submit any number gain $0$ points. (In the case of a tie, all teams that tied gain the same amount of points.) [b]p23.[/b] Ribbotson the Frog is at the point $(0, 0)$ and wants to reach the point $(18, 18)$ in $36$ steps. Each step, he either moves one unit in the $+x$ direction or one unit in the $+y$ direction. However, Ribbotson hates turning, so he must make at least two steps in any direction before switching directions. If $m$ is the number of different paths Ribbotson the Frog can make, estimate $m$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21\left(1 -\left|\log_{10}\frac{N}{m} \right|^2\right)$. [b]p24.[/b] Let $M = \pi^{\pi^{\pi^{\pi}}}$. Estimate $k$, where $M = 10^{10^{k}}$. If $N$ is your team’s submitted number, your team earns points equal to the closest integer to $21 \cdot 1.01^{(-|N-k|^3)}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3248729p29808138]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Jorge writes a list with an even number of integers, not all equal to $0$ (there may be repeated numbers). Show that Martin can cross out a number from the list, of his choice, so that it is impossible for Jorge to separate the remaining numbers into two groups in such a way that the sum of all the numbers in one group is equal to the sum of all the others. numbers from the other group.

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.