Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$. [i]Liubomir Chiriac, Moldova[/i]

We have $ 2n\plus{}1$ elements in the commutative ring $ R$: \[ \alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .\] Let us define the elements \[ \sigma_k\equal{}k\alpha \plus{} \sum_{i\equal{}1}^n \alpha_i\varrho_i^k .\] Prove that the ideal $ (\sigma_0,\sigma_1,...,\sigma_k,...)$ can be finitely generated. [i]L. Redei[/i]

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

Let $C$ and $C'$ be two concentric circles of radii $r$ and $r'$ respectively. Determine how much the quotient $r'/r$ must be worth so that in the limited crown (annulus) through $C$ and $C'$ there are eight circles $C_i$ , $i = 1, . . . , 8$, which are tangent to $C$ and to $C'$ , and also that $C_i$ is tangent to $C_{i+1}$ for $i = 1, . . . ,7$ and $C_8$ tangent to $C_1$ .

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

If $a,b,c>0,a+b+c=1$,then: $\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$

In an alphabet of $n$ letters, is $syllable$ is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered $indecent$. A $word$ is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of indecent syllables for which infinite words do not exist.

Let $k,m,n$ be positive integers such that $n^m$ is divisible by $m^n$, and $m^k$ is divisible by $k^m$. [list=a] [*] Prove that $n^k$ is divisible by $k^n$. [*] Find an example of $k,m,n$ satisfying the above conditions, where all three numbers are distinct and bigger than 1. [/list]

A frog is jumping on $N$ stones which are numbered from $1$ to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than $\tfrac 12$.

Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]

Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

The set A contains exactly $21$ integers. The sum of any $11$ numbers in $A$ is greater than the sum of the remaining numbers. It is known that the set $A$ contain thes number $101$, and the largest number in $A$ is $2014$. Find out the other $19$ numbers in $A$.

Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\]

Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent. [i]Fajar Yuliawan, Bandung[/i]

A positive integer is randomly selected from among the first $2011$ primes. What is the probability that it is even?

Parallelogram $ABCD$ is the base of a pyramid $SABCD$. Planes determined by triangles $ASC$ and $BSD$ are mutually perpendicular. Find the area of the side $ASD$, if areas of sides $ASB,BSC$ and $CSD$ are equal to $x,y$ and $z$, respectively.

Two beads, each of mass $m$, are free to slide on a rigid, vertical hoop of mass $m_h$. The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction. [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(3,0)); draw(circle((1.5,1),1)); filldraw(circle((1.4,1.99499),0.1), gray(.3)); filldraw(circle((1.6,1.99499),0.1), gray(.3)); [/asy]

For a subset $ S$ of vertices of graph $ G$, let $ \Lambda(S)$ be the subset of all edges of $ G$ such that at least one of their ends is in $ S$. Suppose that $ G$ is a graph with $ m$ edges. Let $ d^*: V(G)\longrightarrow\mathbb N\cup\{0\}$ be a function such that a) $ \sum_{u}d^*(u)\equal{}m$. b) For each subset $ S$ of $ V(G)$: \[ \sum_{u\in S}d^*(u)\leq|\Lambda(S)|\] Prove that we can give directions to edges of $ G$ such that for each edge $ e$, $ d^\plus{}(e)\equal{}d^*(e)$.

The number 1987 can be written as a three digit number $xyz$ in some base $b$. If $x + y + z = 1 + 9 + 8 + 7$, determine all possible values of $x$, $y$, $z$, $b$.

Two points $A$ and $B$ and line $\ell$ are fixed in the plane so that $\ell$ is not perpendicular to $AB$ and does not intersect the segment $AB$. We consider all circles with a centre $O$ not lying on $\ell$, passing through $A$ and $B$ and meeting $\ell$ at some points $C$ and $D$. Prove that all the circumcircles of triangles $OCD$ touch a fixed circle.