Let $M$ be an arbitrary point of a circle circumscribed around an acute-angled triangle $ABC$. Prove that the product of the distances from the point $M$ to the sides $AC$ and $BC$ is equal to the product of the distances from $M$ to the side $AB$ and to the tangent to the circumscribed circle at point $C$.

If $a$ and $b$ are positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$, what is the greatest possible value of $a+b$?

For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.

Consider the acute angle $ABC$. On the half-line $BC$ we consider the distinct points $P$ and $Q$ whose projections onto the line $AB$ are the points $M$ and $N$. Knowing that $AP=AQ$ and $AM^2-AN^2=BN^2-BM^2$, find the angle $ABC$.

Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.

Given a natural number $n{}$ find the smallest $\lambda$ such that\[\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,\] for any positive integers $y{}$ and $x \geqslant y + n$.

Find all real numbers $m$ such that $$\frac{1-m}{2m} \in \{x\ |\ m^2x^4+3mx^3+2x^2+x=1\ \forall \ x\in \mathbb{R} \}$$

An infinite sequence $(a_0,a_1,a_2,\dots)$ of positive integers is called a $\emph{ribbon}$ if the sum of any eight consecutive terms is at most $16$; that is, for all $i\ge0$, \[a_i+a_{i+1}+\dots+a_{i+7}\le16.\]A positive integer $m$ is called a $\emph{cut size}$ if every ribbon contains a set of consecutive elements that sum to $m$; that is, given any ribbon $(a_0,a_1,a_2,\dots)$, there exist nonnegative integers $k\le l$ such that \[a_k+a_{k+1}+\dots+a_l=m.\]Find, with proof, all cut sizes, or prove that none exist.

Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.

Show that every subset of {1,2,...,99,100} with 55 elements contains at least 2 numbers with a difference of 9.

The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle{BAC}$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle{PMN}=\angle{MQN}=90^{\circ}$. If $PN=5$ and $BC=3$, then the length of $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $(a+b)$?

Let $n$ be a positive integer. In an $n\times4$ table, each row is equal to \[\begin{tabular}{| c | c | c | c |} \hline 2 & 0 & 1 & 0 \\ \hline \end{tabular}\] A [i]change[/i] is taking three consecutive boxes in the same row with different digits in them and changing the digits in these boxes as follows: \[0\to1\text{, }1\to2\text{, }2\to0\text{.}\] For example, a row $\begin{tabular}{| c | c | c | c |}\hline 2 & 0 & 1 & 0 \\ \hline\end{tabular}$ can be changed to the row $\begin{tabular}{| c | c | c | c |}\hline 0 & 1 & 2 & 0 \\ \hline\end{tabular}$ but not to $\begin{tabular}{| c | c | c | c |}\hline 2 & 1 & 2 & 1 \\ \hline\end{tabular}$ because $0$, $1$, and $0$ are not distinct. Changes can be applied as often as wanted, even to items already changed. Show that for $n<12$, it is not possible to perform a finite number of changes so that the sum of the elements in each column is equal.

In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term in the sequence is 4000. What is the first term? $\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 10$

There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

Let a real-valued sequence $\{x_n\}_{n\geqslant 1}$ be such that $$\lim_{n\to\infty}nx_n=0$$ Find all possible real values of $t$ such that $\lim_{n\to\infty}x_n\big(\log n\big)^t=0$ .

Dora has $8$ rods with lengths $1, 2, 3, 4, 5, 6, 7$ and $8$ cm. Dora chooses $4$ of the rods and uses them to assemble a trapezoid (the $4$ chosen rods must be the $4$ sides). How many different trapezoids can she obtain in this way? Two trapezoids are considered different if they are not congruent.

Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.

Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 < x < y < z <p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x ^ 2 + y ^ 2 + z ^ 2$ is divisible by $x + y + z$.

The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear.

$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$. For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$, we always have $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$. Find the minimum of $\lambda$

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

Several circles are drawn on the plane and all points of their intersection or touching are marked. Is it possible that each circle contains exactly five marked points and each point belongs to exactly five circles?