$|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$. There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$.

p1. The pattern $ABCCCDDDDABBCCCDDDDABBCCCDDDD...$ repeats to infinity. Which letter ranks in place $2533$ ? p2. Prove that if $a > 2$ and $b > 3$ then $ab + 6 > 3a + 2b$. p3. Given a rectangle $ABCD$ with size $16$ cm $\times 25$ cm, $EBFG$ is kite, and the length of $AE = 5$ cm. Determine the length of $EF$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/885af838bcf1392eb02e2764f31ae83cb84b78.png[/img] p4. Consider the following series of statements. It is known that $x = 1$. Since $x = 1$ then $x^2 = 1$. So $x^2 = x$. As a result, $x^2 - 1 = x- 1$ $(x -1) (x + 1) = (x - 1) \cdot 1$ Using the rule out, we get $x + 1 = 1$ $1 + 1 = 1$ $2 = 1$ The question. a. If $2 = 1$, then every natural number must be equal to $ 1$. Prove it. b. The result of $2 = 1$ is something that is impossible. Of course there's something wrong in the argument above? Where is the fault? Why is that you think wrong? p5. To calculate $\sqrt{(1998)(1996)(1994)(1992)+16}$ . someone does it in a simple way as follows: $2000^2-2 \times 5\times 2000 + 5^2 - 5$? Is the way that person can justified? Why? p6. To attract customers, a fast food restaurant give gift coupons to everyone who buys food at the restaurant with a value of more than $25,000$ Rp.. Behind every coupon is written one of the following numbers: $9$, $12$, $42$, $57$, $69$, $21$, 15, $75$, $24$ and $81$. Successful shoppers collect coupons with the sum of the numbers behind the coupon is equal to 100 will be rewarded in the form of TV $21''$. If the restaurant owner provides as much as $10$ $21''$ TV pieces, how many should be handed over to the the customer? p7. Given is the shape of the image below. [img]https://cdn.artofproblemsolving.com/attachments/4/6/5511d3fb67c039ca83f7987a0c90c652b94107.png[/img] The centers of circles $B$, $C$, $D$, and $E$ are placed on the diameter of circle $A$ and the diameter of circle $B$ is the same as the radius of circle $A$. Circles $C$, $D$, and $E$ are equal and the pairs are tangent externally such that the sum of the lengths of the diameters of the three circles is the same with the radius of the circle $A$. What is the ratio of the circumference of the circle $A$ with the sum of the circumferences of circles $B$, $C$, $D$, and $E$? p8. It is known that $a + b + c = 0$. Prove that $a^3 + b^3 + c^3 = 3abc$.

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?

[u]Round 5[/u] [b]5.1.[/b] Julia baked a pie for herself to celebrate pi day this year. If Julia bakes anyone pie on pi day, the following year on pi day she bakes a pie for herself with $1/3$ probability, she bakes her friend a pie with $1/6$ probability, and she doesn't bake anyone a pie with $1/2$ probability. However, if Julia doesn't make pie on pi day, the following year on pi day she bakes a pie for herself with $1/2$ probability, she bakes her friend a pie with $1/3$ probability, and she doesn't bake anyone a pie with $1/6$ probability. The probability that Julia bakes at least $2$ pies on pi day in the next $5$ years can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.2.[/b] Steven is flipping a coin but doesn't want to appear too lucky. If he ips the coin $8$ times, the probability he only gets sequences of consecutive heads or consecutive tails that are of length $4$ or less can be expressed as $p/q$, for relatively prime positive integers $p$ and $q$. Compute $p + q$. [b]5.3.[/b] Let $ABCD$ be a square with side length $3$. Further, let $E$ be a point on side$ AD$, such that $AE = 2$ and $DE = 1$, and let $F$ be the point on side $AB$ such that triangle $CEF$ is right with hypotenuse $CF$. The value $CF^2$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [u]Round 6[/u] [b]6.1.[/b] Let $P$ be a point outside circle $\omega$ with center $O$. Let $A,B$ be points on circle $\omega$ such that $PB$ is a tangent to $\omega$ and $PA = AB$. Let $M$ be the midpoint of $AB$. Given $OM = 1$, $PB = 3$, the value of $AB^2$ can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]6.2.[/b] Let $a_0, a_1, a_2,...$with each term defined as $a_n = 3a_{n-1} + 5a_{n-2}$ and $a_0 = 0$, $a_1 = 1$. Find the remainder when $a_{2020}$ is divided by $360$. [b]6.3.[/b] James and Charles each randomly pick two points on distinct sides of a square, and they each connect their chosen pair of points with a line segment. The probability that the two line segments intersect can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 7[/u] [b]7.1.[/b] For some positive integers $x, y$ let $g = gcd (x, y)$ and $\ell = lcm (2x, y)$: Given that the equation $xy+3g+7\ell = 168$ holds, find the largest possible value of $2x + y$. [b]7.2.[/b] Marco writes the polynomials $$f(x) = nx^4 +2x^3 +3x^2 +4x+5$$ and $$g(x) = a(x-1)^4 +b(x-1)^3 +6(x-1)^2 + d(x - 1) + e,$$ where $n, a, b, d, e$ are real numbers. He notices that $g(i) = f(i) - |i|$ for each integer $i$ satisfying $-5 \le i \le -1$. Then $n^2$ can be expressed as $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]7.3. [/b]Equilateral $\vartriangle ABC$ is inscribed in a circle with center $O$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $BC$, respectively. Segment $\overline{CD}$ intersects $\overline{AB}$ and $\overline{AE}$ at $Y$ and $X$, respectively. Given that $\vartriangle DXE$ and $\vartriangle AXC$ have equal area, $\vartriangle AXY$ has area $ 1$, and $\vartriangle ABC$ has area $52$, find the area of $\vartriangle BXC$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of total webpage visits our website received last month. Let $B$ be the number photos in our photo collection from ABMC onsite 2017. Let $M$ be the mean speed round score. Further, let $C$ be the number of times the letter c appears in our problem bank. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766251p24226451]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Ten math students take a test, and the average score on the test is $28$. If five students had an average of $15$, what was the average of the other five students' scores? [b]p2.[/b] If $a\otimes b = a^2 + b^2 + 2ab$, find $(-5\otimes 7) \otimes 4$. [b]p3.[/b] Below is a $3 \times 4$ grid. Fill each square with either $1$, $2$ or $3$. No two squares that share an edge can have the same number. After filling the grid, what is the $4$-digit number formed by the bottom row? [img]https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png[/img] [b]p4.[/b] What is the angle in degrees between the hour hand and the minute hand when the time is $6:30$? [b]p5.[/b] In a small town, there are some cars, tricycles, and spaceships. (Cars have $4$ wheels, tricycles have $3$ wheels, and spaceships have $6$ wheels.) Among the vehicles, there are $24$ total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town? [b]p6.[/b] You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails? [b]p7.[/b] In the below diagram, $\angle ABC$ and $\angle BCD$ are right angles. If $\overline{AB} = 9$, $\overline{BD} = 13$, and $\overline{CD} = 5$, calculate $\overline{AC}$. [img]https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png[/img] [b]p8.[/b] Out of $100$ customers at a market, $80$ purchased oranges, $60$ purchased apples, and $70$ purchased bananas. What is the least possible number of customers who bought all three items? [b]p9.[/b] Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following: Francis: Fred ate the cookies. Fred: Ted did not eat the cookies. Ted: Francis is lying. If exactly one of them is telling the truth, who ate all the cookies? [b]p11.[/b] Let $ABC$ be a triangle with a right angle at $A$. Suppose $\overline{AB} = 6$ and $\overline{AC} = 8$. If $AD$ is the perpendicular from $A$ to $BC$, what is the length of $AD$? [b]p12.[/b] How many three digit even numbers are there with an even number of even digits? [b]p13.[/b] Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line. [b]p14.[/b] A man $5$ foot, $10$ inches tall casts a $14$ foot shadow. $20$ feet behind the man, a flagpole casts ashadow that has a $9$ foot overlap with the man's shadow. How tall (in inches) is the flagpole? [b]p15.[/b] Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has $n$ balls and 3 divides $n$, then he throws away a third of the balls. If $3$ does not divide $n$ but $2$ divides $n$, then he throws away half of them. If neither $3$ nor $2$ divides $n$, he stops throwing away the balls. If he began with $1458$ balls, after how many steps does he stop throwing away balls? [b]p16.[/b] Oski has $50$ coins that total to a value of $82$ cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter). [b]p17.[/b] Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC$. Suppose $\overline{MA} = \overline{MB} = \overline{MC} = 2$ and $\angle ACB = 30^o$. Find the area of the triangle. [b]p18.[/b] A spirited integer is a positive number representable in the form $20^n + 13k$ for some positive integer $n$ and any integer $k$. Determine how many spirited integers are less than $2013$. [b]p19. [/b]Circles of radii $20$ and $13$ are externally tangent at $T$. The common external tangent touches the circles at $A$, and $B$, respectively where $A \ne B$. The common internal tangent of the circles at $T$ intersects segment $AB$ at $X$. Find the length of $AX$. [b]p20.[/b] A finite set of distinct, nonnegative integers $\{a_1, ... , a_k\}$ is called admissible if the integer function $f(n) = (n + a_1) ... (n + a_k)$ has no common divisor over all terms; that is, $gcd \left(f(1), f(2),... f(n)\right) = 1$ for any integer$ n$. How many admissible sets only have members of value less than $10$? $\{4\}$ and $\{0, 2, 6\}$ are such sets, but $\{4, 9\}$ and $\{1, 3, 5\}$ are not. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n>1$ such that every prime that divides $n^6-1$ also divides $n^5-n^3-n^2+1$.

[i]In this round, problems will depend on the answers to other problems. A bolded letter is used to denote a quantity whose value is determined by another problem's answer.[/i] [u]Part I[/u] [b]p1.[/b] Let F be the answer to problem number $6$. You want to tile a nondegenerate square with side length $F$ with $1\times 2$ rectangles and $1 \times 1$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p2.[/b] Let [b]A[/b] be the answer to problem number $1$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]A[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac{7\sqrt5}{4}$ and $PD = \frac74$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p3.[/b] Let [b]B[/b] be the answer to problem number $2$. Let $S$ be the set of positive integers less than or equal to [b]B[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p4.[/b] Let [b]C[/b] be the answer to problem number $3$. You have $9$ shirts and $9$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as blue pants. Given that you have [b]C[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p5.[/b] Let [b]D[/b] be the answer to problem number $4$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + a = gcd(a, b) + b =$ [b]D[/b]. Find $ab$. [b]p6.[/b] Let [b]E[/b] be the answer to problem number $5$. A function $f$ defined on integers satisfies $f(y)+f(12-y) = 10$ and $f(y) + f(8 - y) = 4$ for all integers $y$. Given that $f($ [b]E[/b] $) = 0$, compute $f(4)$. [u]Part II[/u] [b]p7.[/b] Let [b]L[/b] be the answer to problem number $12$. You want to tile a nondegenerate square with side length [b]L[/b] with $1\times 2$ rectangles and $7\times 7$ squares. The rectangles can be oriented in either direction. How many ways can you do this? [b]p8.[/b] Let [b]G[/b] be the answer to problem number $7$. Triangle $ABC$ has a right angle at $B$ and the length of $AC$ is [b]G[/b]. Let $D$ be the midpoint of $AB$, and let $P$ be a point inside triangle $ABC$ such that $PA = PC = \frac12$ and $PD = \frac{1}{2010}$ . The length of $AB^2$ is expressible as $m/n$ for $m, n$ relatively prime positive integers. Find $m$. [b]p9.[/b] Let [b]H[/b] be the answer to problem number $8$. Let $S$ be the set of positive integers less than or equal to [b]H[/b]. What is the maximum size of a subset of $S$ whose elements are pairwise relatively prime? [b]p10.[/b] Let [b]I[/b] be the answer to problem number $9$. You have $391$ shirts and $391$ pairs of pants. Each is either red or blue, you have more red shirts than blue shirts, and you have same number of red shirts as red pants. Given that you have [b]I[/b] ways of wearing a shirt and pants whose colors match, find out how many red shirts you own. [b]p11.[/b] Let [b]J[/b] be the answer to problem number $10$. You have two odd positive integers $a, b$. It turns out that $lcm(a, b) + 2a = 2 gcd(a, b) + b = $ [b]J[/b]. Find $ab$. [b]p12.[/b] Let [b]K[/b] be the answer to problem number $11$. A function $f$ defined on integers satisfies $f(y)+f(7-y) = 8$ and $f(y) + f(5 - y) = 4$ for all integers $y$. Given that $f($ [b]K[/b] $) = 453$, compute $f(2)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$, then $$ \frac{n^2}{m-n} $$ is also an element of $S$.

[u]Round 1[/u] [b]p1.[/b] Nineteen witches, all of different heights, stand in a circle around a campfire. Each witch says whether she is taller than both of her neighbors, shorter than both, or in-between. Exactly three said “I am taller.” How many said “I am in-between”? [b]p2.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?. [b]p3.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol? [img]https://cdn.artofproblemsolving.com/attachments/a/3/78814b37318adb116466ede7066b0d99d6c64d.png[/img] [b]p4.[/b] A zebra is a new chess piece that jumps in the shape of an “L” to a location three squares away in one direction and two squares away in a perpendicular direction. The picture shows all the moves a zebra can make from its given position. Is it possible for a zebra to make a sequence of $64$ moves on an $8\times 8$ chessboard so that it visits each square exactly once and returns to its starting position? [img]https://cdn.artofproblemsolving.com/attachments/2/d/01a8af0214a2400b279816fc5f6c039320e816.png[/img] [b]p5.[/b] Ann places the integers $1, 2,..., 100$ in a $10 \times 10$ grid, however she wants. In each round, Bob picks a row or column, and Ann sorts it from lowest to highest (left-to-right for rows; top-to-bottom for columns). However, Bob never sees the grid and gets no information from Ann. After eleven rounds, Bob must name a single cell that is guaranteed to contain a number that is at least $30$ and no more than $71$. Can he find a strategy to do this, no matter how Ann originally arranged the numbers? [u]Round 2[/u] [b]p6.[/b] Evelyn and Odette are playing a game with a deck of $101$ cards numbered $1$ through $101$. At the start of the game the deck is split, with Evelyn taking all the even cards and Odette taking all the odd cards. Each shuffles her cards. On every move, each player takes the top card from her deck and places it on a table. The player whose number is higher takes both cards from the table and adds them to the bottom of her deck, first the opponent’s card, then her own. The first player to run out of cards loses. Card $101$ was played against card $2$ on the $10$th move. Prove that this game will never end. [img]https://cdn.artofproblemsolving.com/attachments/8/1/aa16fe1fb4a30d5b9e89ac53bdae0d1bdf20b0.png[/img] [b]p7.[/b] The Vogon spaceship Tempest is descending on planet Earth. It will land on five adjacent buildings within a $10 \times 10$ grid, crushing any teacups on roofs of buildings within a $5 \times 1$ length of blocks (vertically or horizontally). As Commander of the Space Force, you can place any number of teacups on rooftops in advance. When the ship lands, you will hear how many teacups the spaceship breaks, but not where they were. (In the figure, you would hear $4$ cups break.) What is the smallest number of teacups you need to place to ensure you can identify at least one building the spaceship landed on? [img]https://cdn.artofproblemsolving.com/attachments/8/7/2a48592b371bba282303e60b4ff38f42de3551.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true : $\bullet$ $a\ge b$ $\bullet$ $ab - cd = cd -ba$. For example, $2011$ is pora because $20-11 = 11-02$ Find all the numbers around.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

Let $a_1, a_2, ... , a_{2012}$ be odd positive integers. Prove that the number $$A=\sqrt{a^2_1+ a^2_2+ ...+ a^2_{2012}-1}$$ is irrational.

Two digits one are written at the ends of a segment. In the middle, their sum is written, the number 2. Then, in the middle between each two neighboring numbers written, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written? [i]Proposed by G. Halperin[/i]

Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$