An equilateral triangle $ABC$ is inscribed in a square with side 1 (each vertex of the triangle is on a side of the square and no two are on the same side). Determine the greatest and smallest value of the side of $\Delta ABC$.

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]B1 / G1[/b] Find $20^3 + 2^2 + 3^1$. [b]B2[/b] A piece of string of length $10$ is cut $4$ times into strings of equal length. What is the length of each small piece of string? [b]B3 / G2[/b] What is the smallest perfect square that is also a perfect cube? [b]B4[/b] What is the probability a $5$-sided die with sides labeled from $1$ through $5$ rolls an odd number? [b]B5 / G3[/b] Hanfei spent $14$ dollars on chicken nuggets at McDonalds. $4$ nuggets cost $3$ dollars, $6$ nuggets cost $4$ dollars, and $12$ nuggets cost $9$ dollars. How many chicken nuggets did Hanfei buy? [u]Set 2[/u] [b]B6[/b] What is the probability a randomly chosen positive integer less than or equal to $15$ is prime? [b]B7[/b] Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled $1$ through $6$. What is the probability that the sum is greater than $5$? [b]B8 / G4[/b] What is the radius of a circle with area $4$? [b]B9[/b] What is the maximum number of equilateral triangles on a piece of paper that can share the same corner? [b]B10 / G5[/b] Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in $20$ minutes. Bab can make a pizza in $30$ minutes. If Bob and Bab want to make $50$ pizzas in total, how many hours would that take them? [u]Set 3[/u] [b]B11 / G6[/b] Find the area of an equilateral rectangle with perimeter $20$. [b]B12 / G7[/b] What is the minimum possible number of divisors that the sum of two prime numbers greater than $2$ can have? [b]B13 / G8[/b] Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins? [b]B14 / G9[/b] Aven has $4$ distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table? [b]B15 / G10[/b] Find the largest $7$-digit palindrome that is divisible by $11$. PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a square $(p-1)\times(p-1)$, where $p$ is a prime number, which is divided by squares $1\times 1$ whose sides are parallel to the initial square's sides. Show that it is possible to select $p$ vertices such that there are no three collinear vertices.

A rectangle is divided into $9$ smaller rectangles. The area of four of them is $5, 3, 9$ and $2$, as in the picture below. (The picture is not at scale.) [img]https://cdn.artofproblemsolving.com/attachments/8/e/0ccd6f41073f776b62e9ef4522df1f1639ee31.png[/img] Determine the minimum area of the rectangle. Under what circumstances is it achieved?

Find the positive integer less than 18 with the most positive divisors.

a) Let $n$ $(n \geq 2)$ be an integer. On a line there are $n$ distinct (pairwise distinct) sets of points, such that for every integer $k$ $(1 \leq k \leq n)$ the union of every $k$ sets contains exactly $k+1$ points. Show that there is always a point that belongs to every set. b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer $k$ the union of every $k$ sets contains exactly $k+1$ points?

Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

We are given an acute-angled triangle $ABC$ and a random point $X$ in its interior, different from the centre of the circumcircle $k$ of the triangle. The lines $AX,BX$ and $CX$ intersect $k$ for a second time in the points $A_1,B_1$ and $C_1$ respectively. Let $A_2,B_2$ and $C_2$ be the points that are symmetric of $A_1,B_1$ and $C_1$ in respect to $BC,AC$ and $AB$ respectively. Prove that the circumcircle of the triangle $A_2,B_2$ and $C_2$ passes through a constant point that does not depend on the choice of $X$.

Compute $99(99^2+3) + 3\cdot99^2$. [i]Proposed by Evan Chen[/i]

$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]

Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number? [i]Proposed by Andy Xu[/i]

A [i]string of parentheses[/i] is any word that can be composed by the following rules. 1) () is a string of parentheses. 2) If $s$ is a string of parentheses then $(s)$ is a string of parentheses. 3) If $s$ and t are strings of parentheses then $st$ is a string of parentheses. The [i]midcode [/i] of a string of parentheses is the tuple of natural numbers obtained by finding, for all pairs of opening and its corresponding closing parenthesis, the number of characters remaining to the left from the medium position between these parentheses, and writing all these numbers in non-decreasing order. For example, the midcode of $(())$ is $(2,2)$ and the midcode of ()() is $(1,3)$. Prove that midcodes of arbitrary two different strings of parentheses are different.

Two distinct positive even integers sum to $8$. Find the larger of the two integers.

Alice sends a secret message to Bob using her RSA public key $n=400000001$. Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n$. Eve knows that $n=pq$, a product of two prime factors. Find $p$ and $q$.

Let $n$ be an positive integer. We call $n$ $\textit{good}$ when there exists positive integer $k$ s.t. $n=k(k+1)$. Does there exist 2022 pairwise distinct $\textit{good}$ numbers s.t. their sum is also $\textit{good}$ number?

Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.

Let $f_0$ be the function from $\mathbb{Z}^2$ to $\{0,1\}$ such that $f_0(0,0)=1$ and $f_0(x,y)=0$ otherwise. For each positive integer $m$, let $f_m(x,y)$ be the remainder when \[ f_{m-1}(x,y) + \sum_{j=-1}^{1} \sum_{k=-1}^{1} f_{m-1}(x+j,y+k) \] is divided by $2$. Finally, for each nonnegative integer $n$, let $a_n$ denote the number of pairs $(x,y)$ such that $f_n(x,y) = 1$. Find a closed form for $a_n$. [i]Proposed by Bobby Shen[/i]

In triangle $ABC$, $AB=AC$ and $B\hat{A}C=100^\circ$. Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$. Prove that $AD+DB=BC$.

A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.

Prove that if  $(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1$ for real numbers $x,y$, then $x+y = 0$.

Determine the natural numbers a, b, c s.t. : $ \frac{3a+2b}{6a}=\frac{8b+c}{10b}=\frac{3a+2c}{3c} $ and $ a^{2}+b^{2}+c^{2}=975 $ The challenge here is to come up with as basic solution as possible.

Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$.