Note that if the product of any two distinct members of {1,16,27} is increased by 9, the result is the perfect square of an integer. Find the unique positive integer $n$ for which $n+9,16n+9,27n+9$ are also perfect squares.

Let $ n $ be a positive integer. A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$ if for every $ k $ $(1\leq k\leq n)$ the term $ a_k $ is equal with the number of distinct numbers from the subsequence $(a_1,a_2,\ldots,a_k).$ a) How many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ do exist? b) For every positive integer $m$ find how many balanced sequences $(a_1,a_2,\ldots,a_n)$ of length $ n $ exist such that $a_n=m.$

Let $ G$ be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of $ G$ is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.) [i]P. Erdos[/i]

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.

Is it possible to cover the plane with (infinite) circles so that exactly $2020$ circles pass through each point on the plane?

In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\frac{EG}{GF}$ equals [asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label("A", (0,0), S); label("B", (12,0), S); label("C", (14,7.75), E); label("E", (8.75,4.84), N); label("F", (5,0), S); label("M", (13,3.875), E); label("G", (7,1)); [/asy] $ \textbf{(A)}\ \frac{3}{2} \qquad\textbf{(B)}\ \frac{4}{3} \qquad\textbf{(C)}\ \frac{5}{4} \qquad\textbf{(D)}\ \frac{6}{5} \\ \qquad\textbf{(E)}\ \text{not enough information to solve the problem} $

A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color? $\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$

Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have $$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$ For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions: $\textbf{(i)}$ it contains no digits of $9$; $\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square. Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$

In a triangle $ABC$, the perpendicular bisectors of sides $AB$ and $AC$ intersect $BC$ at $X$ and $Y$. Prove that $BC = XY$ if and only if $\tan B\tan C = 3$ or $\tan B\tan C = -1$.

Let $\triangle ABC$ be a triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD= 48 \frac{1}{61}$ and $DC=61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE=11$. Find $AE$.

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$

Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order. We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.

Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation: $|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.

Let $A(n)$ be the least possible number of distinct points in the plane with the following property: For every $k = 1,2,...,n$ there is a line containing precisely $k$ of these points. Show that $A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]$

I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$. $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$. Prove that $\left[AI\right]=\left[ND\right]$.

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$. [i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]

Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r$ of its inscribed circle, and the radius $r_c$ of its ex-circle tangent to the side $AB$ and the extensions of $BC$ and $CA$.

Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.

Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$

The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.

In a very long corridor there is an infinite number of cabinets, which start with $1,2,3,...$ numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set $A \subseteq \{1, 2,3,...\}$ . In ascending order, the QED people now cause chaos: the person with number $a \in A$ visits the cabinet with the numbers $a,2a,3a,...$ opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from $A$ open.