Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.

Dertemine integers $a_{1},a_{2},...,a_{99}=a_{0}$ satisfying $|a_{k}-a_{k-1}|\geq 1996$ for all $k=1,2,...,99$, such that $m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|$ is minimum possible, and find the minimum value $m^{*}$ of $m$.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be externally tangent at a point $A$. A line tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $C$ and $D$; then the segment $AB$ is extended to intersect $\mathcal{C}_2$ at a point $E$. Let $F$ be the midpoint of $\overarc{CD}$ that does not contain $E$, and let $H$ be the intersection of $BF$ with $\mathcal{C}_2$. Show that $CD$, $AF$, and $EH$ are concurrent.

Let $p > 3$ be a prime number. A sequence of $p-1$ integers $a_1,a_2, \dots, a_{p-1}$ is called [i]wonky[/i] if they are distinct modulo \(p\) and $a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p$ for all \(i \in \{1, 2, \dots, p-1\}\), where \(a_p = a_1\) and \(a_{p+1} = a_2\). Does there always exist a wonky sequence such that $$a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,$$ are all distinct modulo $p$? [i]Proposed by Harun Khan[/i]

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour. [i]Note: Any rectangle is assumed here to have sides contained in the lines of the grid.[/i] [i](Bulgaria - Nikolay Beluhov)[/i]

A set of $103$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $101$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)

Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$

Determine all strictly increasing functions $ f: R\rightarrow R$ satisfying relationship $ f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005$ for any real values of x and y.

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. Rule 1: If the integer is less than 10, multiply it by 9. Rule 2: If the integer is even and greater than 9, divide it by 2. Rule 3: If the integer is odd and greater than 9, subtract 5 from it. Find the $98th$ term of the sequence that begins $ 98, 49,\ldots . $ $ \text{(A)}\ 6\qquad\text{(B)}\ 11\qquad\text{(C)}\ 22\qquad\text{(D)}\ 27\qquad\text{(E)}\ 54 $

In triangle $ABC, \angle BAC = 60^o, \angle ACB = 90^o$ and $D$ is on $BC$. If $AD$ bisects $\angle BAC$ and $CD = 3$ cm, calculate $DB$ .

Let $ ABC$ be an acute-angled triangle and let $ D,E,F$ be the feet of perpendiculars from $ A,B,C$ respectively to $ BC,CA,AB .$ Let the perpendiculars from $ F$ to $ CB,CA,AD,BE$ meet them in $ P,Q,M,N$ respectively. Prove that the points $ P,Q,M,N$ are collinear.

From the set $ \{1,2,3,\ldots,2009\}$ we choose $ 1005$ numbers, such that sum of any $ 2$ numbers isn't neither $ 2009$ nor $ 2010$. Find all ways on we can choose these $ 1005$ numbers.

An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?

Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre

8 chess players participated in the chess tournament and everyone played exactly one game with everyone else. It is known that any two chess players who play a draw with each other ended up scoring different numbers of points. Find the greatest possible number of draws in this tournament. (For winning the game the chess player is awarded $1$ point, for a draw $1/2$ points, for defeat $0$ points.)

How many perfect squares have the digits $1$ through $9$ each exactly once when written in base $10$? You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, your score will be $\lfloor12.5\cdot\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor.$

Prove that if triangle $T_1$ contains triangle $T_2$, then the perimeter of triangle $T_1$ is not less than the perimeter of triangle $T_2$.

Let $ p $ be prime. Denote by $ N (p) $ the number of integers $ x $ for which $ 1 \leq x \leq p $ and $$ x ^ {x} \equiv 1 \quad (\bmod p) $$Prove that there exist numbers $ c <1/2 $ and $ p_ {0}> 0 $ such that $$ N (p) \leq p ^ {c} $$if $ p \ge p_ {0} $.

Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove $$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Equilateral triangle $DEF$ is inscribed inside equilateral triangle $ABC$ such that $DE$ is perpendicular to $BC$. Let $x$ be the area of triangle $ABC$ and $y$ be the area of triangle $DEF$. Compute $\tfrac{x}{y}$.

Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1. [i]Ankan Bhattacharya[/i]

Find all positive integer pairs $(a,b)$ for which the set of positive integers can be partitioned into sets $H_1$ and $H_2$ such that neither $a$ nor $b$ can be represented as the difference of two numbers in $H_i$ for $i=1,2$.