$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.

[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$. [b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$. [b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$. [b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$. [b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$? [b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$. Find $P\left( \frac{1}{2010}\right)$. [b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$. [b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers. [b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ , For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ . Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

Give $P(x) = {x^{2022}} + {a_{2021}}{x^{2021}} + ... + {a_1}x + 1$ is a polynomial with real coefficents. a) Assume that $2021a_{2021}^2 - 4044{a_{2020}} < 0.$ Prove that: $P(x)$ can't have $2022$ real roots. b) Assume that $a_1^2 + a_2^2 + ... + a_{2021}^2 \le \frac{4}{{2021}}.$ Prove that: $P(x)\ge 0$, for all $x\in \mathbb{R}.$

Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and: $$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$ Find the value of $a_{2022!}$.

Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.

Let $f(x)=x^2-2$ and let $f^{(n)}(x)$ denote the $n$-th iteration of $f$. Let $H=\{x:f^{(100)}(x)\leq -1\}$. Find the length of $H$ (the sum of the lengths of the intervals of $H$).

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

Let $(A,+,\cdot)$ be a finite unit ring, with $n\geq 3$ elements in which there exist [b]exactly[/b] $\dfrac {n+1}2$ perfect squares (e.g. a number $b\in A$ is called a perfect square if and only if there exists an $a\in A$ such that $b=a^2$). Prove that a) $1+1$ is invertible; b) $(A,+,\cdot)$ is a field. [i]Proposed by Marian Andronache[/i]

Find all functions $f: (0, \infty) \to (0, \infty)$ such that \begin{align*} f(y(f(x))^3 + x) = x^3f(y) + f(x) \end{align*} for all $x, y>0$. [i]Proposed by Jason Prodromidis, Greece[/i]

A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction? [b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race? [b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.). [img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img] [b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$. [b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle. [b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

[b]p1.[/b] Find the smallest positive integer which is $1$ more than multiple of $3$, $2$ more than a multiple of $4$, and $4$ more than a multiple of $7$. [b]p2.[/b] Let $p = 4$, and let $a =\sqrt1$, $b =\sqrt2$, $c =\sqrt3$, $...$. Compute the value of $(p-a)(p-b) ... (p-z)$. [b]p3.[/b] There are $6$ points on the circumference of a circle. How many convex polygons are there having vertices on these points? [b]p4.[/b] David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident? [b]p5.[/b] A pyramid is created by stacking equilateral triangles of balls, each layer having one fewer ball per side than the triangle immediately beneath it. How many balls are used if the pyramid’s base has $5$ balls to a side? [b]p6.[/b] Call a positive integer $n$ good if it has $3$ digits which add to $4$ and if it can be written in the form $n = k^2$, where $k$ is also a positive integer. Compute the average of all good numbers. [b]p7.[/b] John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake? [b]p8.[/b] Evaluate $\sum^{10}_{i=1}\sum^{10}_{j=1} ij$. [b]p9.[/b] If three numbers $a$, $b$, and $c$ are randomly selected from the interval $[-2, 2]$, what is the probability that $a^2 + b^2 + c^2 \ge 4$? [b]p10.[/b] Evaluate $\sum^{\infty}_{x=2} \frac{2}{x^2 - 1}.$ [b]p11.[/b] Consider $4x^2 - kx - 1 = 0$. If the roots of this polynomial are $\sin \theta$ and $\cos \theta$, compute $|k|$. [b]p12.[/b] Given that $65537 = 2^{16} + 1$ is a prime number, compute the number of primes of the form $2^n + 1$ (for $n \ge 0$) between $1$ and $10^6$. [b]p13.[/b] Compute $\sin^{-1}(36/85) + \cos^{-1}(4/5) + \cos^{-1}(15/17).$ [b]p14.[/b] Find the number of integers $n$, $1\le n \le 2003$, such that $n^{2003} - 1$ is a multiple of $10$. [b]p15.[/b] Find the number of integers $n,$ $1 \le n \le 120$, such that $n^2$ leaves remainder $1$ when divided by $120$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each positive integer $k$, we define the polynomial $S_k(x)=1+x+x^2+x^3+...+x^{k-1}$ Show that $n \choose 1$ $S_1(x) +$ $n \choose 2$ $S_2(x) +$ $n \choose 3$ $S_3(x)+...+$ $n \choose n$ $S_n(x) = 2^{n-1}S_n\left(\frac{1+x}{2}\right)$ for every positive integer $n$ and every real number $x$.

Given a positive real number $t$ , determine the sets $A$ of real numbers containing $t$ , for which there exists a set $B$ of real numbers depending on $A$ , $|B| \geq 4$ , such that the elements of the set $AB =\{ ab \mid a\in A , b \in B \}$ form a finite arithmetic progression .

Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ : $$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.