Richard comes across an infinite row of magic hats, $H_1, H_2, \dots$ each of which may contain a dollar bill with probabilities $p_1, p_2, \dots$. If Richard draws a dollar bill from $H_i$, then $p_{i+1} = p_i$, and if not, $p_{i+1}=\frac{1}{2}p_i$. If $p_1 = \frac{1}{2}$ and $E$ is the expected amount of money Richard makes from all the hats, compute $\lfloor 100E \rfloor$. [i]Proposed by Alex Li[/i]

Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that: [list] [*]The leftmost number is 1. [*]The rightmost number is 10. [*]Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list] How many such permutations are there?

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.

Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]

If $x=\frac{1}{3}$, what is the value, rounded to $100$ decimal digits, of $\sum_{n=0}^{7}\frac{2^n}{1+x^{2^n}}$?

Find all primes $p$ such that there exist an integer $n$ and positive integers $k, m$ which satisfies the following. $$ \frac{(mk^2+2)p-(m^2+2k^2)}{mp+2} = n^2$$

Let $n$ be a positive integer and let $S \subseteq \{0, 1\}^n$ be a set of binary strings of length $n$. Given an odd number $x_1, \dots, x_{2k + 1} \in S$ of binary strings (not necessarily distinct), their [i]majority[/i] is defined as the binary string $y \in \{0, 1\}^n$ for which the $i^{\text{th}}$ bit of $y$ is the most common bit among the $i^{\text{th}}$ bits of $x_1, \dots,x_{2k + 1}$. (For example, if $n = 4$ the majority of 0000, 0000, 1101, 1100, 0101 is 0100.) Suppose that for some positive integer $k$, $S$ has the property $P_k$ that the majority of any $2k + 1$ binary strings in $S$ (possibly with repetition) is also in $S$. Prove that $S$ has the same property $P_k$ for all positive integers $k$. [i]Proposed by Joshua Brakensiek[/i]

The Young’s modulus, $E$, of a material measures how stiff it is; the larger the value of $E$, the more stiff the material. Consider a solid, rectangular steel beam which is anchored horizontally to the wall at one end and allowed to deflect under its own weight. The beam has length $L$, vertical thickness $h$, width $w$, mass density $\rho$, and Young’s modulus $E$; the acceleration due to gravity is $g$. What is the distance through which the other end moves? ([i]Hint: you are expected to solve this problem by eliminating implausible answers. All of the choices are dimensionally correct.[/i]) (a) $h \exp\left( \frac{\rho gL}{E} \right)$ (b) $2\frac{\rho gh^2}{E}$ (c) $\sqrt{2Lh}$ (d) $\frac{3}{2}\frac{\rho gL^4}{Eh^2}$ (e) $\sqrt{3}\frac{EL}{\rho gh}$

Richard rolls a fair six-sided die repeatedly until he rolls his twentieth prime number or his second even number. Compute the probability that his last roll is prime.

Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$.

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation $ x^2\plus{} p_1x\plus{}q_1 \equal{} 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 \plus{} p_2x \plus{} q_2 \equal{} 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.

Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

In the acute triangle $ABC$ the side $BC> AB$, and the angle bisector $BL = AB$. On the segment $BL$ there is a point $M$, for which $\angle AML = \angle BCA$. Prove that $AM = LC$.

A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares? [asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy] $ \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{5}2 $

Let $m,n$ be two positive integers with $m \ge n \ge 2022$. Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be $2n$ real numbers. Prove that the numbers of ordered pairs $(i,j) ~(1 \le i,j \le n)$ such that \[ |a_i+b_j-ij| \le m \] does not exceed $3n\sqrt{m \log n}$.

In Wonderland there are at least $5$ towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most $2$ gold coins.

What are the positive integer numbers that we are able to obtain in $2007$ distinct ways, when the sum is at least out of two positive consecutive integers? What is the smallest of all of them? Example: the number 9 is written in exactly two such distinct ways: $9 = 4 + 5$ $9 = 2 + 3 + 4.$

Given a set $S \subset R^+$, $S \ne \emptyset$ such that for all $a, b, c \in S$ (not necessarily distinct) then $a^3 + b^3 + c^3 - 3abc$ is rational number. Prove that for all $a, b \in S$ then $\frac{a - b}{a + b}$ is also rational.

Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$

For any $a,b,c>0$ satisfying $a+b+c+ab+ac+bc= 3$, prove that \[2\leq a+b+c+abc\leq 3\] [i]Proposed by Mohammad Ahmadi[/i]

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets. (a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket. (b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket. [b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties. (a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$. (b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$. [b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer. [b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$. [b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that: (a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle. (b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

Prove that the equation $(3x+4y)(4x+5y)=7^z$ doesn't have solution in natural numbers.