Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence $a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence $ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.

For $p\in (0;\infty)$ find the area of the region bounded by the curves $y^{2}=4px$ and $16py^{2}=5(x-p)^{3}$

[b]p1.[/b] There are $4$ mirrors facing the inside of a $5\times 7$ rectangle as shown in the figure. A ray of light comes into the inside of a rectangle through $A$ with an angle of $45^o$. When it hits the sides of the rectangle, it bounces off at the same angle, as shown in the diagram. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, $D$? A bounce is a time when the ray hit a mirror and reflects off it. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d6ea83941cdb4b2dab187d09a0c45782af1691.png[/img] [b]p2.[/b] Jerry cuts $4$ unit squares out from the corners of a $45\times 45$ square and folds it into a $43\times 43\times 1$ tray. He then divides the bottom of the tray into a $43\times 43$ grid and drops a unit cube, which lands in precisely one of the squares on the grid with uniform probability. Suppose that the average number of sides of the cube that are in contact with the tray is given by $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p3.[/b] Compute $2021^4 - 4 \cdot 2023^4 + 6 \cdot 2025^4 - 4 \cdot 2027^4 + 2029^4$. [b]p4.[/b] Find the number of distinct subsets $S \subseteq \{1, 2,..., 20\}$, such that the sum of elements in $S$ leaves a remainder of $10$ when divided by $32$. [b]p5.[/b] Some $k$ consecutive integers have the sum $45$. What is the maximum value of $k$? [b]p6.[/b] Jerry picks $4$ distinct diagonals from a regular nonagon (a regular polygon with $9$-sides). A diagonal is a segment connecting two vertices of the nonagon that is not a side. Let the probability that no two of these diagonals are parallel be $\frac{m}{n}$ where $m, n$ are positive integers that are relatively prime. Find $m + n$. [b]p7.[/b] The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure [img]https://cdn.artofproblemsolving.com/attachments/1/7/9dafe6b72aa8471234afbaf4c51e3e97c49ee5.png[/img] Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$. [b]p8.[/b] Let $P(x)$ be an integer polynomial (polynomial with integer coefficients) with $P(-5) = 3$ and $P(5) = 23$. Find the minimum possible value of $|P(-2) + P(2)|$. [b]p9. [/b]There exists a unique tuple of rational numbers $(a, b, c)$ such that the equation $$a \log 10 + b \log 12 + c \log 90 = \log 2025.$$ What is the value of $a + b + c$? [b]p10.[/b] Each grid of a board $7\times 7$ is filled with a natural number smaller than $7$ such that the number in the grid at the $i$th row and $j$th column is congruent to $i + j$ modulo $7$. Now, we can choose any two different columns or two different rows, and swap them. How many different boards can we obtain from a finite number of swaps? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(a) Show that for any angle $\theta$ and for any natural number $m$: $$| \sin m\theta| \le m| \sin \theta|$$ (b) Show that for all angles $\theta_1$ and $\theta_2$ and for all even natural numbers $m$: $$| \sin m \theta_2 - \sin m \theta_1| \le m| \sin (\theta_2 - \theta_1)|$$ (c) Show that for every odd natural number $m$ there are two angles, resp. $\theta_1$ and $\theta_2$, exist for which the inequality in (b) is not valid.

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

Find all functions $f\colon R \to R$ such that \[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\] for all reals $x,y$.

Find all real numbers $x$ which satisfy $\frac{n}{3n+1}\leq x\leq \frac{4n+1}{2n-1}$, for all $n\in\mathbb{N}^*$. [i]Gheorghe Boroica[/i]

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

Let $F(x_1,..., x_n)$ be a polynomial with real coefficients in $ n > 1$ “indeterminate” variables $x_1,..., x_n$. We say that $F$ is $n$-[i]alternating [/i]if for all integers $1 \le i < j \le n$, $$F(x_1,..., x_i,..., x_j,..., x_n) = - F(x_1,..., x_j,..., x_i,..., x_n),$$ i.e. swapping the order of indeterminates $x_i, x_j$ flips the sign of the polynomial. For example, $x^2_1x_2 - x^2_2x_1$ is $2$-alternating, whereas $x_1x_2x_3 +2x_2x_3$ is not $3$-alternating. [i]Note: two polynomials $P(x_1,..., x_n)$ and $Q(x_1,..., x_n)$ are considered equal if and only if each monomial constituent $ax^{k_1}_1... x^{k_n}_n$ of $P$ appears in $Q$ with the same coefficient $a$, and vice versa. This is equivalent to saying that $P(x_1,..., x_n) = 0$ if and only if every possible monomial constituent of $P$ has coefficient $0$. [/i] (1) Compute a $3$-alternating polynomial of degree $3$. (2) Prove that the degree of any nonzero $n$-alternating polynomial is at least ${n \choose 2}$.

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function such that $f(y) - f(x) < y - x$ for all real numbers $x$ and $y > x$. The sequence $u_1,u_2,\ldots$ of real numbers is such that $u_{n+2} = f(u_{n+1}) - f(u_n)$ for all $n\geq 1$. Prove that for any $\varepsilon > 0$ there exists a positive integer $N$ such that $|u_n| < \varepsilon$ for all $n\geq N$.

Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a^2}{b+c} + \frac{2b^2}{c+a} + \frac{2c^2}{a+b} \geq a+b+c\](this is, of course, a joke!) [b]EDITED with exponent 2 over c[/b]

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

Given $n$ non-negative real numbers, $n\geq 3$, such that the sum of the $n$ numbers is less than or equal to $3$ and the sum of the squares of the $n$ numbers is greater than or equal to $1$, prove that among the $n$ numbers three can be chosen whose sum is greater than or equal to $1$.

Solve the system of equations: $$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\ \log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$

Let $S = \{1, 2, \cdots , n\}, n \in N$. Find the number of functions $f : S \to S$ with the property that $x + f(f(f(f(x)))) = n + 1$ for all $x \in S$?

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it true that if $H$ and $A$ are bounded subsets of $\mathbb{R}$, then there exists at most one set $B$ such that $a+b(a\in A,b\in B)$ are pairwise distinct and $H=A+B$.

Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.

Let $a_1$, $a_2$, $\ldots$, $a_n$ be a geometric progression with $a_1 = \sqrt{2}$ and $a_2 = \sqrt[3]{3}$. What is \[\displaystyle{\frac{a_1+a_{2013}}{a_7+a_{2019}}}?\]

Let $\eta \in [0, 1]$ be a relative measure of material absorbence. $\eta$ values for materials combined together are additive. $\eta$ for a napkin is $10$ times that of a sheet of paper, and a cardboard roll has $\eta = 0.75$. Justin can create a makeshift cup with $\eta = 1$ using $50$ napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with $\eta = 1$?

The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$ \[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]