[b]p1.[/b] Consider a parallelogram $ABCD$ with sides of length $a$ and $b$, where $a \ne b$. The four points of intersection of the bisectors of the interior angles of the parallelogram form a rectangle $EFGH$. A possible configuration is given below. Show that $$\frac{Area(ABCD)}{Area(EFGH)}=\frac{2ab}{(a - b)^2}$$ [img]https://cdn.artofproblemsolving.com/attachments/e/a/afaf345f2ef7c8ecf4388918756f0b56ff20ef.png[/img] [b]p2.[/b] A metal wire of length $4\ell$ inches (where $\ell$ is a positive integer) is used as edges to make a cardboard rectangular box with surface area $32$ square inches and volume $8$ cubic inches. Suppose that the whole wire is used. (i) Find the dimension of the box if $\ell= 9$, i.e., find the length, the width, and the height of the box without distinguishing the different orders of the numbers. Justify your answer. (ii) Show that it is impossible to construct such a box if $\ell = 10$. [b]p3.[/b] A Pythagorean n-tuple is an ordered collection of counting numbers $(x_1, x_2,..., x_{n-1}, x_n)$ satisfying the equation $$x^2_1+ x^2_2+ ...+ x^2_{n-1} = x^2_{n}.$$ For example, $(3, 4, 5)$ is an ordinary Pythagorean $3$-tuple (triple) and $(1, 2, 2, 3)$ is a Pythagorean $4$-tuple. (a) Given a Pythagorean triple $(a, b, c)$ show that the $4$-tuple $(a^2, ab, bc, c^2)$ is Pythagorean. (b) Extending part (a) or using any other method, come up with a procedure that generates Pythagorean $5$-tuples from Pythagorean $3$- and/or $4$-tuples. Few numerical examples will not suffice. You have to find a method that will generate infinitely many such $5$-tuples. (c) Find a procedure to generate Pythagorean $6$-tuples from Pythagorean $3$- and/or $4$- and/or $5$-tuples. Note. You can assume without proof that there are infinitely many Pythagorean triples. [b]p4.[/b] Consider the recursive sequence defined by $x_1 = a$, $x_2 = b$ and $$x_{n+2} =\frac{x_{n+1} + x_n - 1}{x_n - 1}, n \ge 1 .$$ We call the pair $(a, b)$ the seed for this sequence. If both $a$ and $b$ are integers, we will call it an integer seed. (a) Start with the integer seed $(2, 2019)$ and find $x_7$. (b) Show that there are infinitely many integer seeds for which $x_{2020} = 2020$. (c) Show that there are no integer seeds for which $x_{2019} = 2019$. [b]p5.[/b] Suppose there are eight people at a party. Each person has a certain amount of money. The eight people decide to play a game. Let $A_i$, for $i = 1$ to $8$, be the amount of money person $i$ has in his/her pocket at the beginning of the game. A computer picks a person at random. The chosen person is eliminated from the game and their money is put into a pot. Also magically the amount of money in the pockets of the remaining players goes up by the dollar amount in the chosen person's pocket. We continue this process and at the end of the seventh stage emerges a single person and a pot containing $M$ dollars. What is the expected value of $M$? The remaining player gets the pot and the money in his/her pocket. What is the expected value of what he/she takes home? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.

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

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

Consider the sequence $a_1, a_2, a_3, \ldots$ with $$ a_n = \frac{1}{n(n+1)}.$$ In how many ways can the number $\frac{1}{1980}$ be represented as the sum of finitely many consecutive terms of this sequence?

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

Find a four-digit number whose division by two given distinct one-digit numbers goes along the following lines: [img]https://cdn.artofproblemsolving.com/attachments/2/a/e1d3c68ec52e11ad59de755c3dbdc2cf54a81f.png[/img]

The Fibonacci sequence is given by $a_1 = 1, a_2 = 2$ and $a_{n+1} = a_n +a_{n-1}$ for $n > 1$. Express $a_{2n}$ in terms of only $a_{n-1},a_n,a_{n+1}$.

Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$. (AY Belov,)

The sequence $\{a_n\}_{n\ge 1}$ of real numbers is defined as follows: $$a_1=1, \quad \text{and}\quad a_{n+1}=\frac{1}{2\lfloor a_n \rfloor -a_n+1} \quad \text{for all} \quad n\ge 1$$ Find $a_{2024}$.

Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and \[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \] for all integers $n \ge 1$. Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$. [i]Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.[/i]

2. Prove that for any $2\le n \in \mathbb{N}$ there exists positive integers $a_1,a_2,\cdots,a_n$ such that $\forall i\neq j: \text{gcd}(a_i,a_j) = 1$ and $\forall i: a_i \ge 1402$ and the given relation holds. $$[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]$$

A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .

Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.

Let $a\geq 4$ be a positive integer. Determine the smallest value of $n\geq 5$, $n\neq a$, such that $a$ can be represented in the form$$a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}$$for a suitable choice of the $n$ positive integers $x_1,x_2,\ldots ,x_n$.

[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps? [b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. (a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back. (b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.) [b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. [b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].