If the natural numbers $a, b, c, d$ verify the relationships: $$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$ $$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$ and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.

Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers. Determine if there are infinitely many pairs of triangles that are pals of each other.

A set of different positive integers is called meaningful if for any finite nonempty subset the corresponding arithmetic and geometric means are both integers. $a)$ Does there exist a meaningful set which consists of $2019$ numbers? $b)$ Does there exist an infinite meaningful set? Note: The geometric mean of the non-negative numbers $a_1, a_2,\cdots, $ $a_n$ is defined as $\sqrt[n]{a_1a_2\cdots a_n} .$

Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.

Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).

p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$. p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ? p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements. p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of ​​the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$. p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.

Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number \[ \frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}} \] is an integer?

$f(n)$ denotes the largest integer $k$ such that that $2^k|n$. $2006$ integers $a_i$ are such that $a_1<a_2<...<a_{2016}$. Is it possible to find integers $k$ where $1 \le k\le 2006$ and $f(a_i-a_j)\ne k$ for every $1 \le j \le i \le 2006$ ?

Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer. (V. Senderov, Moscow)

Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?

Choose nine of the digits from $0$ to $9$ and place them in the boxes in the figure so that there are no repeated digits and the indicated sum is correct. [img]https://cdn.artofproblemsolving.com/attachments/6/2/7f06575ec70eb9ddd58c6cf9dd3cb60d306e7c.png[/img] Which digit was not used? You can fill in the boxes so that the unused digit is other?

A positive integer number is written in red on each side of a square. The product of the two red numbers on the adjacent sides is written in green for each corner point. The sum of the green numbers is $40$. Which values ​​are possible for the sum of the red numbers? (G. Kirchner, University of Innsbruck)

Given the $6$ digits: $0, 1, 2, 3, 4, 5$. Find the sum of all even four-digit numbers which can be expressed with the help of these figures (the same figure can be repeated).

Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \] [b]OR[/b] Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]

Let be a natural number $ n. $ Prove that there exists a number $ k\in\{ 0,1,2,\ldots n \} $ such that the floor of $ 2^{n+k}\sqrt 2 $ is even.

For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.

A finite set of positive integers is called [i]isolated [/i]if the sum of the numbers in any given proper subset is co-prime with the sum of the elements of the set. a) Prove that the set $A=\{4,9,16,25,36,49\}$ is isolated; b) Determine the composite numbers $n$ for which there exist the positive integers $a,b$ such that the set \[ A=\{(a+b)^2, (a+2b)^2,\ldots, (a+nb)^2\}\] is isolated.

Let $m>1$ be a natural number and $N=m^{2017}+1$. On a blackboard, left to right, are written the following numbers: \[N, N-m, N-2m,\dots, 2m+1,m+1, 1.\] On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] While computing $7 - 2002 \cdot x$, John accidentally evaluates from left to right $((7 - 2002) \cdot x)$ instead of correctly using order of operations $(7 - (2002 \cdot x))$. If he gets the correct answer anyway, what is $x$? [b]p2.[/b] Given that $$x^2 + y^2 + z^2 = 6$$ $$ \left( \frac{x}{y} + \frac{y}{x} \right)^2 + \left( \frac{y}{z} + \frac{z}{y} \right)^2 + \left( \frac{z}{x} + \frac{x}{z} \right)^2 = 16.5,$$ what is $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}$ ? [b]p3.[/b] Evaluate $$\frac{tan \frac{\pi}{4}}{4}+\frac{tan \frac{3\pi}{4}}{8}+\frac{tan \frac{5\pi}{4}}{16}+\frac{tan \frac{7\pi}{4}}{32}+ ...$$ [b]p4.[/b] Note that $2002 = 22 \cdot 91$, and so $2002$ is a multiple of the number obtained by removing its middle $2$ digits. Generalizing this, how many $4$-digit palindromes, $abba$, are divisible by the $2$-digit palindrome, $aa$? [b]p5.[/b] Let $ABCDE$ be a pyramid such that $BCDE$ is a square with side length $2$, and $A$ is $2$ units above the center of $BCDE$. If $F$ is the midpoint of $\overline{DE}$ and $G$ is the midpoint of $\overline{AC}$, what is the length of $\overline{DE}$? [b]p6.[/b] Suppose $a_1, a_2,..., a_{100}$ are real numbers with the property that $$i(a_1 + a_2 +... + a_i) = 1 + (a_{i+1} + a_{i+2} + ... + a_{100})$$ for all $i$. Compute $a_{10}$. [b]p7.[/b] A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance nit needs to travel along the block’s surface to reach the opposite corner? [b]p8.[/b] In the number game, a pair of positive integers $(n,m)$ is written on a blackboard. Two players then take turns doing the following: 1. If $n \ge m$, the player chooses a positive integer $c$ such that $n - cm \ge 0$, and replaces $(n,m)$ with $(n - cm,m)$. 2. If $m > n$, the player chooses a positive integer $c$ such that $m - cn \ge 0$, and replaces $(n,m)$ with $(n,m - cn)$. If $m$ or $n$ ever become $0$, the game ends, and the last player to have moved is declared the winner. If $(n,m)$ are originally $(20021000, 2002)$, what choices of $c$ are winning moves for the first player? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that: a) There is a sequence of non-zero natural numbers $a_1, a_2, ...$ uniquely determined, so that: $n = \sum _ {d | n} a _ d$ for whatever $n \in N ^ {*}$ . b) There is a sequence of non-zero natural numbers $b_1, b_2, ...$ uniquely determined, so that: $n = \prod _ {d | n} b _ d$ for whatever $n \in N ^ {*}$ . Note: The sum from a), respectively the product from b), are made after all the natural divisors $d$ of the number $n$ , including $1$ and $n$ .

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

Find all integers $x,y$ such that $2x+3y=185$ and $xy>x+y$.

Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]