Let $ T$ be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$. 2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$. For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list] [*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or [*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down). [/list] Thus, for any $k$, the ant can choose to go to one of eight possible points. \\ Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

The function $F (x)$ is defined on $R$ and has a second derivative for each value of the variable. Prove that there is a point $x_0$ such that the product $ F(x_0) F''(x_0)$ is non-negative. PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source[/url], it is not clear if it means $ F(x_0) F''(x_0)$ or $ F(x_0) F'(x_0)$.

Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$, then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.

Find all positive integers $n\geq 2$, for which there exist positive integers $a_1, a_2, ..., a_n$ such that both sets $$\{a_1, a_2, ..., a_n\}\;\;\;and\;\;\;\{a_1+a_2, a_2+a_3, ..., a_n+a_1\}$$ contain $n$ consecutive integers.

If positive real numbers $x,y,z$ satisfies $x+y+z=3,$ prove that $$\sum_{\text{cyc}} y^2z^2<3+\sum_{\text{cyc}} yz.$$ [i]Proposed by Li4 and Untro368.[/i]

In a tree with $n$ vertices, for each vertex $x_i$, denote the longest paths passing through it by $l_i^1,l_i^2,...,l_i^{k_i}$. $x_i$ cuts those longest paths into two parts with $(a_i^1,b_i^1),(a_i^2,b_i^2),...,(a_i^{k_i},b_i^{k_i})$ vertices respectively. If $\max_{j=1,...,k_i} \{a_i^j\times b_i^j\}=p_i$, find the maximum and minimum values of $\sum_{i=1}^{n} p_i$. [i]Proposed by Sina Rezaei[/i]

A [i]Pattano coin[/i] is a coin which has a blue side and a yellow side. A positive integer not exceeding $100$ is written on each side of every coin (the sides may have different integers). Two Pattano coins are [i]identical [/i] if the number on the blue side of both coins are equal and the number on the yellow side of both coins are equal. Two Pattano coins are [i]pairable [/i] if the number on the blue side of both coins are equal or the number on the yellow side of both coins are equal. Given $2559$ Pattano coins such that no two coins are identical. Show that at least one Pattano coin is pairable with at least $50$ other coins

Let $f : N \to R$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$

For a constant $a,$ let $f(x)=ax\sin x+x+\frac{\pi}{2}.$ Find the range of $a$ such that $\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).$

[asy] import olympiad; import cse5; size(5cm); pointpen = black; pair A = Drawing((10,17.32)); pair B = Drawing((0,0)); pair C = Drawing((20,0)); draw(A--B--C--cycle); pair X = 0.85*A + 0.15*B; pair Y = 0.82*A + 0.18*C; pair W = (-11,0) + X; pair Z = (19, 9); draw(W--X, EndArrow); draw(X--Y, EndArrow); draw(Y--Z, EndArrow); anglepen=black; anglefontpen=black; MarkAngle("\theta", C,Y,Z, 3); [/asy] The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number. [i](Ahaan Rungta, 5 points)[/i]

Let $ABC$ be a scalene acute triangle with incenter $I$ and circumcircle $\Omega$. $M$ is the midpoint of small arc $BC$ on$\Omega$ and $N$ is the projection of $I$ onto the line passing through the midpoints of $AB$ and $AC$. A circle $\omega$ with center $Q$ is internally tangent to $\Omega$ at $A$, and touches segment $BC$. If the circle with diameter $IM$ meets $\Omega$ again at $J$, prove that $JI$ bisects $\angle QJN$. [i]Proposed by David Anghel[/i]

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

All of the digits of a seven-digit positive integer are either $7$ or $8.$ If this integer is divisible by $9,$ what is the sum of its digits?

Different positive $a, b, c$ are such that $a^{239} = ac- 1$ and $b^{239} = bc- 1$.Prove that $238^2 (ab)^{239} <1$.

Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.

Let $f_n$ be $n$th Fibonacci number defined by recurrence $f_{n+1} - f_n - f_{n-1} = 0$, $n \in \mathbb{N}$ and initial conditions $f_0 = 0$, $f_1 = 1$. Prove that for any $n \in \mathbb{N}$ $$(n - 1) (n + 1) (2nf_{n+1} - (n + 6) f_n)$$is divisible by 150 for any $n \in \mathbb{N}$.

Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \][i]Proposed by David Stoner[/i]

Let $\mathcal{P}$ be a convex polygon and $\textbf{T}$ be a triangle with vertices among the vertices of $\mathcal{P}$. By removing $\textbf{T}$ from $\mathcal{P}$, we end up with $0, 1, 2,$ or $3$ smaller polygons (possibly with shared vertices) which we call the effect of $\textbf{T}$. A triangulation of $P$ is a way of dissecting it into some triangles using some non-intersecting diagonals. We call a triangulation of $\mathcal{P}$ $\underline{\text{beautiful}}$, if for each of its triangles, the effect of this triangle contains exactly one polygon with an odd number of vertices. Prove that a triangulation of $\mathcal{P}$ is beautiful if and only if we can remove some of its diagonals and end up with all regions as quadrilaterals.

Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

Given the function $f(x) =|4 - 4|x||- 2$. How many solutions does the equation $f(f(x)) = x$ have?

It is initially considered a number of three different digits, none of which is equal to zero. Changing instead two of its digits meet a second number less than the first. If the difference between the first and second is a two-digit number and the sum of the first and the second is a palindromic number less than $500$, what are the palindromics that can be obtained?

A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false. I. The digit is 1. II. The digit is not 2. III. The digit is 3. IV. The digit is not 4. Which one of the following must necessarily be correct? $ \textbf{(A)}\ \text{I is true.} \qquad \textbf{(B)}\ \text{I is false.}\qquad \textbf{(C)}\ \text{II is true.} \qquad \textbf{(D)}\ \text{III is true.} \qquad \textbf{(E)}\ \text{IV is false.}$

Let there be a unit square initially tiled with four congruent shaded equilateral triangles, as seen below. The total area of all of the shaded regions can be expressed in the form $\frac{a-b\sqrt{c}}{d}$ , where $a, b, c$, and $d$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/b/b/34883cf73da568ca237a13fbc2e0fb9322c2e5.png[/img]