$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$ [i](K. Ivanov )[/i]

Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$. [i]Raja Oktovin, Pekanbaru[/i]

On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles. (A Shapovalov)

A [i]magic[/i] octagon is an octagon whose sides follow the lines of the checkerboard's checkers and the side lengths are $1, 2, 3, 4, 5, 6, 7, 8$ (in any order). What is the largest possible area of the magic octagon? [hide=original wording]Burvju astoņstūris ar astoņstūris, kura malas iet pa rūtiņu lapas rūtiņu līnijām un malu garumi ir 1, 2,3, 4, 5, 6, 7, 8 (jebkādā secībā). Kāds ir lielākais iespējamais burvju astoņstūra laukums?[/hide]

Two $10 \times 24$ rectangles are inscribed in a circle as shown. Find the shaded area. [img]https://cdn.artofproblemsolving.com/attachments/1/7/c97fb0e6f45a52fa751777da6ebc519839e379.png[/img]

A piece of paper is folded in half. A second fold is made at an angle $ \phi$ ($ 0^\circ < \phi < 90^\circ$) to the first, and a cut is made as shown below. [img]12881[/img] When the piece of paper is unfolded, the resulting hole is a polygon. Let $ O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $ O$, and also that $ \angle XOY \equal{} 144^\circ$, where $ X$ and $ Y$ are the the vertices of the hole adjacent to $ O$. Find the value(s) of $ \phi$ (in degrees).

$29$ quadratic polynomials $f_1(x), \ldots, f_{29}(x)$ and $15$ real numbers $x_1<x_2<\ldots<x_{15}$ are given. Prove that for some two given polynomials $f_i(x)$ and $f_j(x)$ the following inequality holds: $$\sum_{k=1}^{14} (f_i(x_{k+1})-f_i(x_k))(f_j(x_{k+1})-f_j(x_k))>0$$ [i]A. Voidelevich[/i]

Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that $$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$

Compute $$\int_0^{\frac{\pi}{2}}\frac{e^x(\sin x+\cos x-2)}{(\cos x-2)^2}dx.$$

$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.

A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? [asy] size(200); defaultpen(linewidth(0.8)); draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300)); draw((82,-204) -- (133,-490) -- (342, -662)); draw((652,-626) -- (600,-727)); draw((447,-458) -- (652,-626) -- (859,-410)); draw((133,-490) -- (184, -388)); draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

A teacher gave a test to a class in which $ 10\%$ of the students are juniors and $ 90\%$ are seniors. The average score on the test was $ 84$. The juniors all received the same score, and the average score of the seniors was $ 83$. What score did each of the juniors receive on the test? $ \textbf{(A)}\ 85 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 93 \qquad \textbf{(D)}\ 94 \qquad \textbf{(E)}\ 98$

In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$. Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$. Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.

Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that : i) triangle $BDZ$ is isosceles ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$ iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$

Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent? [asy] unitsize(12); fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray); draw((3,0)--(3,9)--(2,9)--(2,0)); draw((2,7)--(1,7)--(1,0)); draw((2,8)--(3,8)); draw((2,7)--(3,7)); for (int a = 1; a <= 6; ++a) { draw((1,a)--(3,a)); } fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray); draw((6,0)--(6,3)--(5,3)--(5,0)); draw((5,3)--(5,5)--(4,5)--(4,0)); draw((4,4)--(5,4)); draw((4,3)--(5,3)); draw((4,2)--(6,2)); draw((4,1)--(6,1)); fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray); draw((9,0)--(9,6)--(8,6)--(8,0)); draw((8,6)--(8,9)--(7,9)--(7,0)); draw((7,8)--(8,8)); draw((7,7)--(8,7)); draw((7,6)--(8,6)); for (int a = 1; a <= 5; ++a) { draw((7,a)--(9,a)); } fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray); draw((12,0)--(12,12)--(11,12)--(11,0)); draw((11,9)--(10,9)--(10,0)); draw((11,11)--(12,11)); draw((11,10)--(12,10)); draw((11,9)--(12,9)); for (int a = 1; a <= 8; ++a) { draw((10,a)--(12,a)); } fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray); draw((15,0)--(15,10)--(14,10)--(14,0)); draw((14,8)--(13,8)--(13,0)); draw((14,9)--(15,9)); draw((14,8)--(15,8)); for (int a = 1; a <= 7; ++a) { draw((13,a)--(15,a)); } draw((16,0)--(0,0)--(0,13),black); label("Jan",(2,0),S); label("Feb",(5,0),S); label("Mar",(8,0),S); label("Apr",(11,0),S); label("May",(14,0),S); label("$\textbf{MONTHLY SALES}$",(8,14),N); label("S",(0,8),W); label("A",(0,7),W); label("L",(0,6),W); label("E",(0,5),W); label("S",(0,4),W); draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle); label("Drums",(4,13),W); fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray); draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle); label("Bugles",(15,13),W);[/asy] $\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}$

There is an ant on every vertex of a unit cube. At the time zero, ants start to move across the edges with the velocity of one unit per minute. If an ant reaches a vertex, it alternatively turns right and left (for the first time it will turn in a random direction). If two or more ants meet anywhere on the cube, they die! We know an ant survives after three minutes. Prove that there exists an ant that never dies!

Consider a scalene triangle $ABC$ with incentre $I$ and excentres $I_a,I_b,$ and $I_c$, opposite the vertices $A,B,$ and $C$ respectively. The incircle touches $BC,CA,$ and $AB$ at $E,F,$ and $G$ respectively. Prove that the circles $IEI_a,IFI_b,$ and $IGI_c$ have a common point other than $I$.

[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$. [b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals. [b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. [b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$. [b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors. [i]Proposed by [b]FedeX333X[/b][/i]

A shop sells golf balls, golf clubs and golf hats. Golf balls can be purchased at a rate of $25$ cents for two balls. Golf hats cost $\$1$ each. Golf clubs cost $\$10$ each. At this shop, Ross purchased $100$ items for a total cost of exactly $\$100$ (Ross purchased at least one of each type of item). How many golf hats did Ross purchase?

The angle $B$ of a triangle $ABC$ is $120^o$. Let $M$ be a point on the side $AC$ and $K$ a point on the extension of the side $AB$, such that $BM$ is the internal bisector of the angle $\angle ABC$ and $CK$ is the external bisector corresponding to the angle $\angle ACB$ . The segment $MK$ intersects $BC$ at point $P$. Prove that $\angle APM = 30^o$.

A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. [img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]

In $\Delta ABC$, Let the Incircle touch $\overline{BC}, \overline{CA}, \overline{AB}$ at $X,Y,Z$. Let $I_A,I_B,I_C$ be $A$,$B$,$C-$excenters, respectively. Prove that Incenter of $\Delta ABC$, orthocenter of $\Delta XYZ$ and centroid of $\Delta I_AI_BI_C$ are collinear.