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)

Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?

Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$. Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$. Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$

Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \] we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. [i]Radu Gologan[/i]

[b]p1.[/b] Solve the equation $\log_x (x + 2) = 2$. [b]p2.[/b] Solve the inequality: $0.5^{|x|} > 0.5^{x^2}$. [b]p3.[/b] The integers from 1 to 2015 are written on the blackboard. Two randomly chosen numbers are erased and replaced by their difference giving a sequence with one less number. This process is repeated until there is only one number remaining. Is the remaining number even or odd? Justify your answer. [b]p4.[/b] Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer. [b]p5.[/b] Prove that for any finite sequence of digits there exists an integer the square of which begins with that sequence. [b]p6.[/b] The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$. PS. You should use hide for answers.

In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?

Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.

What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ? $ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $

Prove that a positive integer can be expressed in the form $3x^2+y^2$ iff it can also be expressed in form $u^2+uv+v^2$, where $x,y,u,v$ are all positive integers.

Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that \[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]

Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that \[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\] for all real numbers $a,b,c,d$. [i]Proposed by usjl[/i]

[b]p1.[/b] Calculate $A \cdot B +M \cdot C$, where $A = 1$, $B = 2$, $C = 3$, $M = 13$. [b]p2.[/b] What is the remainder of $\frac{2022\cdot2023}{10}$ ? [b]p3.[/b] Daniel and Bryan are rolling fair $7$-sided dice. If the probability that the sum of the numbers that Daniel and Bryan roll is greater than $11$ can be represented as the fraction $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers, what is $a + b$? [b]p4.[/b] Billy can swim the breaststroke at $25$ meters per minute, the butterfly at $30$ meters per minute, and the front crawl at $40$ meters per minute. One day, he swam without stopping or slowing down, swimming $1130$ meters. If he swam the butterfly for twice as long as the breaststroke, plus one additional minute, and the front crawl for three times as long as the butterfly, minus eight minutes, for how many minutes did he swim? [b]p5.[/b] Elon Musk is walking around the circumference of Mars trying to find aliens. If the radius of Mars is $3396.2$ km and Elon Musk is $73$ inches tall, the difference in distance traveled between the top of his head and the bottom of his feet in inches can be expressed as $a\pi$ for an integer $a$. Find $a$. ($1$ yard is exactly $0.9144$ meters). [b]p6.[/b] Lukas is picking balls out of his five baskets labeled $1$,$2$,$3$,$4$,$5$. Each basket has $27$ balls, each labeled with the number of its respective basket. What is the least number of times Lukas must take one ball out of a random basket to guarantee that he has chosen at least $5$ balls labeled ”$1$”? If there are no balls in a chosen basket, Lukas will choose another random basket. [b]p7.[/b] Given $35_a = 42_b$, where positive integers $a$, $b$ are bases, find the minimum possible value of the sum $a + b$ in base $10$. [b]p8.[/b] Jason is playing golf. If he misses a shot, he has a $50$ percent chance of slamming his club into the ground. If a club is slammed into the ground, there is an $80$ percent chance that it breaks. Jason has a $40$ percent chance of hitting each shot. Given Jason must successfully hit five shots to win a prize, what is the expected number of clubs Jason will break before he wins a prize? [b]p9.[/b] Circle $O$ with radius $1$ is rolling around the inside of a rectangle with side lengths $5$ and $6$. Given the total area swept out by the circle can be represented as $a + b\pi$ for positive integers $a$, $b$ find $a + b$. [b]p10.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD$. [b]p11.[/b] Raymond is eating huge burgers. He has been trained in the art of burger consumption, so he can eat one every minute. There are $100$ burgers to start with. However, at the end of every $20$ minutes, one of Raymond’s friends comes over and starts making burgers. Raymond starts with $1$ friend. If each of his friends makes $1$ burger every $20$ minutes, after how long in minutes will there be $0$ burgers left for the first time? [b]p12.[/b] Find the number of pairs of positive integers $(a, b)$ and $b\le a \le 2022$ such that $a\cdot lcm(a, b) = b \cdot gcd(a, b)^2$. [b]p13.[/b] Triangle $ABC$ has sides $AB = 6$, $BC = 10$, and $CA = 14$. If a point $D$ is placed on the opposite side of $AC$ from $B$ such that $\vartriangle ADC$ is equilateral, find the length of $BD$. [b]p14.[/b] If the product of all real solutions to the equation $(x-1)(x-2)(x-4)(x-5)(x-7)(x-8) = -x^2+9x-64$ can be written as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, b, d) = 1$ and $c$ is squarefree, compute $a + b + c + d$. [b]p15.[/b] Joe has a calculator with the keys $1, 2, 3, 4, 5, 6, 7, 8, 9,+,-$. However, Joe is blind. If he presses $4$ keys at random, and the expected value of the result can be written as $\frac{x}{11^4}$ , compute the last $3$ digits of $x$ when $x$ divided by $1000$. (If there are consecutive signs, they are interpreted as the sign obtained when multiplying the two signs values together, e.g $3$,$+$,$-$,$-$, $2$ would return $3 + (-(-(2))) = 3 + 2 = 5$. Also, if a sign is pressed last, it is ignored.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A store had $376$ chocolate bars. Min bought some of the bars, and Max bought $41$ more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.

The battery life on a computer decreases at a rate proportional to the display brightness. Austin starts off his day with both his battery life and brightness at $100\%$. Whenever his battery life (expressed as a percentage) reaches a multiple of $25$, he also decreases the brightness of his display to that multiple of $25$. If left at $100\%$ brightness, the computer runs out of battery in $1$ hour. Compute the amount of time, in minutes, it takes for Austin’s computer to reach $0\%$ battery using his modified scheme.

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Find all real solutions of the equation \[ \left(1+x^2\right)\left(1+x^3\right)\left(1+x^5\right)=8x^5 \]

Consider an infinite sequence of reals $x_1, x_2, x_3, ...$ such that $x_1 = 1$, $x_2 =\frac{2\sqrt3}{3}$ and with the recursive relationship $$n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.$$ Find $x_{2019}$.

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.

Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$

Let $n\geqslant 2$ be a fixed integer. Consider $n$ real numbers $a_1,a_2,\ldots,a_n$ not all equal and let\[d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.\]Determine in terms of $n{}$ the smalest and largest values the quotient $s/d$ may achieve. [i]Selected from the Kvant Magazine[/i]

For any real numbers $a$ and $b>0$, define an [i]extension[/i] of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$. We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$. Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$, there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$, their extensions covers $A$.

Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.