For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers. [i]S. Berlov[/i]

From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?

Point $A(2,0)$. $P(\sin(2t-\frac{\pi}{3}),\cos(2t-\frac{\pi}{3}))$ is a moving point. When $t$ changes from $\frac{\pi}{12}$ to $\frac{\pi}{4}$, area swept by segment $AP$ is________.

Let $a,b\in\mathbb{R}_+$ such that $a+b=1$. Find the minimum value of the following expression: \[E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.\]

Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circle $S$ through $A,B$, and $O$ intersects $AC$ and $BC$ again at points $P$ and $Q$ respectively. Prove that $CO \perp PQ$.

Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?

Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) } \text{Infinitely many} \qquad $

Let $a,b,c$ be nonnegative real numbers. Prove that \[ a(a - b)(a - 2b) + b(b - c)(b - 2c) + c(c - a)(c - 2a) \geq 0.\][i]Wenyu Cao[/i]

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

[THE PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL] The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level. We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements: 1. The top level is painted with only one color. 2. In the second level, the $3$ doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors. 3. For the first level, we apply the same rules as for the second level. [img]https://cdn.artofproblemsolving.com/attachments/2/3/18ee062b79693c4ccc26bf922a7f54e9f352ee.png[/img] (a) In how many ways the first level can be painted? (b) In how many ways the whole tower model can be painted?

On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.

Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make? Proposed by USJL, ST.

Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i < j \leq 101} x_i x_j$ ?

In a multiple-choice test, there are 4 problems, each having 3 possible answers. In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?

Consider the maximum value of circular cone inscribed to a sphere, the ratio of it to the volume of the sphere is________.

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Define $\tilde{f}:[0,1]\rightarrow\mathbb{R}$ as $$\tilde{f}(x)=\begin{dcases} \frac{1}{x}\cdot\int_0^x f(t)dt &\text{ for } x>0;\\ f(0) &\text{ for }x=0.\end{dcases}$$ Show that: [list=a] [*] $\tilde{f}$ is continuous in $0$ and differentiable on $(0,1]$. [*] the following equality takes place: $$\int_0^1 f^2(x)dx = \left(\int_0^1 f(x)dx\right)^2 + \int_0^1 \left(f(x)-\tilde{f}(x)\right)^2dx.$$

In the expansion of $(ax+b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a+b.$

Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.

It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}$$ where $c=f(1)-1$ $a)$ Solve given functional equation for $A=\mathbb{N}$ $b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$

You want to hang a weight $P$ so that it is $7$ m below a ceiling. To do this, it is suspended by means of a vertical cable attached to the midpoint $M$ of a chain hung by its ends from two points on the ceiling $A$ and $B$ distant from each other $4$ m. The price of the cable $PM$ is $p$ pta/m and that of the chain $AMB$ is $q$ pta/m. It is requested: a) Determine the lengths of the cable and the chain to obtain the lowest price cost of installation. b) Discuss the solution for the different values of the relation $p/q$ of both prices. (It is assumed that the weight is large enough to be considered rectile lines the chain segments $AM$ and $MB$).

For every nonempty subset $R$ of $S = \{1,2,...,10\}$, we define the alternating sum $A(R)$ as follows: If $r_1,r_2,...,r_k$ are the elements of $R$ in the increasing order, then $A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1$. (a) Is it possible to partition $S$ into two sets having the same alternating sum? (b) Determine the sum $\sum_{R} A(R)$, where $R$ runs over all nonempty subsets of $S$.