An object has the shape of a square and has side length $a$. Light beams are shone on the object from a big machine. If $m$ is the mass of the object, $P$ is the power $\emph{per unit area}$ of the photons, $c$ is the speed of light, and $g$ is the acceleration of gravity, prove that the minimum value of $P$ such that the bar levitates due to the light beams is \[ P = \dfrac {4cmg}{5a^2}. \] [i]Problem proposed by Trung Phan[/i]

Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the rst ball drawn is white and the second is black.

Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties: [list] [*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and [*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$. [/list] [i]Viktor Simjanoski, Macedonia[/i]

[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number. [b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s? [b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball? [b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse? [b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many dierent orders can Michael get dressed? [b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number? [b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$) [b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$? [b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes? [b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later). After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? [i]I. Izmest’ev[/i]

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that \[A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.\]

$a+b+c \leq 3000000$ and $a\neq b \neq c \neq a$ and $a,b,c$ are naturals. Find maximum $GCD(ab+1,ac+1,bc+1)$

$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$

Determine if there exists a finite set $S$ formed by positive prime numbers so that for each integer $n\geq2$, the number $2^2 + 3^2 +...+ n^2$ is a multiple of some element of $S$.

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.) Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?

Let $ f$ be a function from the real numbers to the real numbers such that $ f(1) \equal{} 1, f(a\plus{}b) \equal{} f(a)\plus{}f(b)$ for all $ a, b,$ and $ f(x)f \left( \frac{1}{x} \right) \equal{} 1$ for all $ x \neq 0.$ Prove that $ f(x) \equal{} x$ for all real numbers $ x.$

Let $ x_1, \ldots, x_n$ be $ n>1$ real numbers satisfying $ A\equal{}\left |\sum^n_{i\equal{}1}x_i\right |\not \equal{}0$ and $ B\equal{}\max_{1\leq i<j\leq n}|x_j\minus{}x_i|\not \equal{}0$. Prove that for any $ n$ vectors $ \vec{\alpha_i}$ in the plane, there exists a permutation $ (k_1, \ldots, k_n)$ of the numbers $ (1, \ldots, n)$ such that \[ \left |\sum_{i\equal{}1}^nx_{k_i}\vec{\alpha_i}\right | \geq \dfrac{AB}{2A\plus{}B}\max_{1\leq i\leq n}|\alpha_i|.\]

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the smallest three-digit number such that the following holds: If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.

The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station? $ \begin{tabular}{|c|c|c|c|} \hline & Listen & Don't Listen & Total\\ \hline Males & ? & 26 & ?\\ \hline Females & 58 & ? & 96\\ \hline Total & 136 & 64 & 200\\ \hline \end{tabular}$ $ \textbf{(A)}\ 39 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 52 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 75$

It's given a right-angled triangle $ABC (\angle{C}=90^{\circ})$ and area $S$. Let $S_1$ be the area of the circle with diameter $AB$ and $k=\frac{S_1}{S}$\\ a) Compute the angles of $ABC$, if $k=2\pi$ b) Prove it is not possible for k to be $3$

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.

Let $f$ and $g$ be two polynomials such that $f(g(x))=g(f(x))$. If $g(x)$ is linear but not identically equal to $x$, and $f(x)=x^3+60x^2+1000x+c$ for some $c$, find the value of $c$. [i]Clarification[/i]: $g$ is not constant.

In a trapezoid, the two non parallel sides and a base have length $1$, while the other base and both the diagonals have length $a$. Find the value of $a$.

Consider a sequence $a_0$, $a_1$, $\ldots$, $a_9$ of distinct positive integers such that $a_0=1$, $a_i < 512$ for all $i$, and for every $1 \le k \le 9$ there exists $0 \le m \le k-1$ such that \[(a_k-2a_m)(a_k-2a_m-1) = 0.\] Let $N$ be the number of these sequences. Find the remainder when $N$ is divided by $1000$. [i]Based on a proposal by Gyumin Roh[/i]

If $n\geqslant 3$ is an integer and $a_1,a_2,\dotsc ,a_n$ are non-zero integers such that $$a_1a_2\cdots a_n\left( \frac{1}{a_1^2}+\frac{1}{a_2^2} +\cdots +\frac{1}{a_n^2}\right)$$is an integer, does it follow that the product $a_1a_2\cdots a_n$ is divisible by each $a_i^2$?

Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)