We want to cover a rectangular $5 \times 137$ with the following figures, prove that this is impossible. \[\text{Squars are the same and all are } \Huge{1 \times 1}\] [asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt)); label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle); [/asy]

Let \( ABC \) be a triangle with \( AB < AC \). Let \( M \) be the midpoint of \( AC \), and let \( D \) be a point on segment \( AC \) such that \( DB = DC \). Let \( E \) be the point of intersection, different from \( B \), of the circumcircle of triangle \( ABM \) and line \( BD \). Define \( P \) and \( Q \) as the points of intersection of line \( BC \) with \( EM \) and \( AE \), respectively. Prove that \( P \) is the midpoint of \( BQ \).

In the quadrilateral $ABCD$ is $AD \parallel BC$ and the angles $\angle A$ and $\angle D$ are acute. The diagonals intersect in $P$. The circumscribed circles of $\vartriangle ABP$ and $\vartriangle CDP$ intersect the line $AD$ again at $S$ and $T$ respectively. Call $M$ the midpoint of $[ST]$. Prove that $\vartriangle BCM$ is isosceles. [img]https://1.bp.blogspot.com/-C5MqC0RTqwY/Xy1fAavi_aI/AAAAAAAAMSM/2MXMlwb13McCYTrOHm1ZzWc0nkaR1J6zQCLcBGAsYHQ/s0/flanders%2B2016%2Bp1.png[/img]

Let $g(k)$ be the number of partitions of a $k$-element set $M$, i.e., the number of families $\{ A_1,A_2,\ldots ,A_s\}$ of nonempty subsets of $M$ such that $A_i\cap A_j=\emptyset$ for $i\not= j$ and $\bigcup_{i=1}^n A_i=M$. Prove that, for every $n$, \[n^n\le g(2n)\le (2n)^{2n}\]

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

Consider $v,w$ two distinct non-zero complex numbers. Prove that \[|zw+\bar{w}|\le |zv+\bar{v}|,\] for any $z\in\mathbb{C},|z|=1$, if and only if there exists $k\in [-1,1]$ such that $w=kv$. [i]Dan Marinescu[/i]

In the popular game of Minesweeper, some fields of an $a \times b$ board are marked with a mine and on all the remaining fields the number of adjacent fields that contain a mine is recorded. Two fields are considered adjacent if they share a common vertex. For which $k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ is it possible for some $a$ and $b$ , $ab > 2021$, to create a board whose fields are covered in mines, except for $2021$ fields who are all marked with $k$?

[b]p1.[/b] If $x + z = v$, $w + z = 2v$, $z - w = 2y$, and $y \ne 0$, compute the value of $$\left(x + y +\frac{x}{y} \right)^{101}.$$ [b]p2. [/b]Every minute, a snail picks one cardinal direction (either north, south, east, or west) with equal probability and moves one inch in that direction. What is the probability that after four minutes the snail is more than three inches away from where it started? [b]p3.[/b] What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices? [b]p4.[/b] Let $ABCD$ be a rectangle where $AB = 4$ and $BC = 3$. Inscribe circles within triangles $ABC$ and $ACD$. What is the distance between the centers of these two circles? [b]p5.[/b] $C$ is a circle centered at the origin that is tangent to the line $x - y\sqrt3 = 4$. Find the radius of $C$. [b]p6.[/b] I have a fair $100$-sided die that has the numbers $ 1$ through $100$ on its sides. What is the probability that if I roll this die three times that the number on the first roll will be greater than or equal to the sum of the two numbers on the second and third rolls? [b]p7. [/b] List all solutions $(x, y, z)$ of the following system of equations with x, y, and z positive real numbers: $$x^2 + y^2 = 16$$ $$x^2 + z^2 = 4 + xz$$ $$y^2 + z^2 = 4 + yz\sqrt3$$ [b]p8.[/b] $A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon ($7$ sided-figure) centered at the origin where $A_1 = (\sqrt[91]{6}, 0)$. $B_1B_2B_3... B_{13}$ is a regular triskaidecagon ($13$ sided-figure) centered at the origin where $B_1 =(0,\sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$ , where $i$ ranges between $1$ and $7$, inclusive, and $j$ ranges between $1$ and $13$, inclusive. [b]p9.[/b] How many three-digit integers are there such that one digit of the integer is exactly two times a digit of the integer that is in a different place than the first? (For example, $100$, $122$, and $124$ should be included in the count, but $42$ and $130$ should not.) [b]p10.[/b] Let $\alpha$ and $\beta$ be the solutions of the quadratic equation $$x^2 - 1154x + 1 = 0.$$ Find $\sqrt[4]{\alpha}+\sqrt[4]{\beta}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is $\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$

Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.

We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$? $ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$

Prove that among the vertices of any convex nonagon, three can be found forming an obtuse triangle, none of whose sides coincide with the sides of the nonagon. [i] Proposed by A. Yuran [/i]

Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively. Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$. [i]Proposed by Evan Chen[/i]

Prove that for any integer $n$, there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$.

A [i]triangulation[/i] $ \mathcal{T}$ of a polygon $ P$ is a finite collection of triangles whose union is $ P,$ and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of $ P$ is a side of exactly one triangle in $ \mathcal{T}.$ Say that $ \mathcal{T}$ is [i]admissible[/i] if every internal vertex is shared by $ 6$ or more triangles. For example [asy] size(100); dot(dir(-100)^^dir(230)^^dir(160)^^dir(100)^^dir(50)^^dir(5)^^dir(-55)); draw(dir(-100)--dir(230)--dir(160)--dir(100)--dir(50)--dir(5)--dir(-55)--cycle); pair A = (0,-0.25); dot(A); draw(A--dir(-100)^^A--dir(230)^^A--dir(160)^^A--dir(100)^^A--dir(5)^^A--dir(-55)^^dir(5)--dir(100)); [/asy] Prove that there is an integer $ M_n,$ depending only on $ n,$ such that any admissible triangulation of a polygon $ P$ with $ n$ sides has at most $ M_n$ triangles.

What is the sum of the infinite series $\frac{20}{3} +\frac{17}{9} + \frac{20}{27} + \frac{17}{81} + \frac{20}{243} + \frac{17}{729} + ...$ ?

Divide a regular hexagon into $24$ identical small equilateral triangles as shown. To each of the $19$ points that are vertices of at least one of the equilateral triangles, a different number is assigned. Find the maximum possible number of small equilateral triangles with their vertices having numbers in ascending order going clockwise. [asy] size(5.5cm); int n=6; pair[] V= sequence(new pair(int i){return dir(360*i/n);}, n); V.cyclic=true; for(int i=0;i<n;++i){ draw(V[i]--V[i+1],black+0.7bp); draw(V[i]--V[i+3],black+0.7bp); draw(midpoint(V[i]--V[i+1])--midpoint(V[i+2]--V[i+3]),black+0.7bp); } [/asy]

Let $ABCD$ be a trapezoid with $AD \parallel BC$, $AD < BC$. Let $E$ be a point on the side $AB$ and $F$ be point on the side $CD$. The circle $(AEF)$ intersects the segment $AD$ again at $A_1$ and the circle $(CEF)$ intersects these segment $BC$ again at $C_1$. Prove that the lines $A_1 C_1$, $BD$ and $EF$ are concurrent. [i]Proposed by A. Kuznetsov[/i]

Determine the integers $x$ and $y$ for which $\sqrt{4^x + 5^y}$ is rational.

There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$