A $4\times 8$ grid is divided into $32$ unit squares. There are square tiles of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. The goal is to completely cover the grid using exactly $n$ of these tiles. [list=a] [*]Is it possible to do this if $n = 19$? [*]Is it possible to do this if $n = 14$? [*]Is it possible to do this if $n = 7$? [/list] [b]Note:[/b] The tiles cannot overlap or extend beyond the grid.

Let $m \ge 2$ be an integer. Find the smallest integer $n>m$ such that for any partition of the set $\{m,m+1,\cdots,n\}$ into two subsets, at least one subset contains three numbers $a, b, c$ such that $c=a^{b}$.

$a,b,c,d\in\mathbb{R}$, solve the system of equations: \[ \begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases} \]

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

An L-shaped figure composed of $4$ unit squares (such as shown in the picture) we call L-dominoes. [img]https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png[/img] Determine the maximum number of L-dominoes that can be placed on a board of dimensions $n \times n$, where $n$ is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page). Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency. Nguyễn Văn Linh

The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is: $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$

Let $n$ be an integer greater than or equal to $2$. Consider real numbers $a_1, a_2, \dots, a_{2n}$ satisfying the condition \[ |a_k - a_{n+k}| \geqslant 1 \quad \text{for all } 1 \leqslant k \leqslant n. \] Determine the minimum possible value of \[ (a_1 - a_2)^2 + (a_2 - a_3)^2 + \dots + (a_{2n-1} - a_{2n})^2 + (a_{2n} - a_1)^2. \]

Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$. (a) Prove that the points $P,Q,R,S$ lie on a circle. (b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.

Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$. [i]Proposed by Ali Khezeli[/i]

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

Ella the Witch was mixing a magic elixir which consisted of three components: $140$ ml of reindeer moss tea, $160$ ml of fly agaric extract, and $50$ ml of moonshine. She took an empty $350$ ml bottle, poured $140$ ml of reindeer moss tea into it and started adding fly agaric extract when she was disturbed by its black cat Mehsto. So she mistakenly poured too much fly agaric extract into the bottle and noticed her fault only later when the bottle Riled before all $50$ ml of moonshine was added. Ella made quick calculations, carefully shaked up the contents of the bottle, poured out some part of liquid and added some amount of mixture of reindeer moss tea and fly agaric extract taken in a certain proportion until the bottle was full again and the elixir had exactly the right compositsion. Which was the proportion of reindeer moss tea and fly agaric extract in the mixture that Ella added into the bottle?