In the triangle $ABC$ the length of side $AB$, and height $AH$ are known. also we know that $\angle B = 2 \angle C.$ Plot this triangle.

Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.

Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.

Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.

In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.

A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G>B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$ and both player gains $1/2$ point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can $B$ take? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests?

Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression Here $d$ is an integer.

There are $2n$ points on a circle, $n$ are red and $n$ are blue. Janson found a red frog and a blue frog at a red point and a blue point on the circle respectively. Every minute, the red frog moves to the next red point in the clockwise direction and the blue frog moves to the next blue point in the anticlockwise direction. Prove that for any initial position of the two frogs, Janson can draw a line through the circle, such that the two frogs are always on opposite sides of the line.

There is a board with $2020$ squares in the bottom row and $2019$ in the top row, located as shown shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/f/3/516ad5485c399427638c3d1783593d79d83002.png[/img] In the bottom row the integers numbers from $ 1$ to $2020$ are placed in some order. Then in each box in the top row records the multiplication of the two numbers below it. How can they place the numbers in the bottom row so that the sum of the numbers in the top row be the smallest possible?

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$

Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.

There are $5$ people left in a game of Among Us, $4$ of whom are crewmates and the last is the impostor. None of the crewmates know who the impostor is. The person with the most votes is ejected, unless there is a tie in which case no one is ejected. Each of the $5$ remaining players randomly votes for someone other than themselves. The probability the impostor is ejected can be expressed as $\frac{m}{n}$. Find $m+n$. [i]Proposed by Sammy Charney[/i]

Coin $ A$ is flipped three times and coin $ B$ is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same? $ \textbf{(A)}\ \frac {19}{128}\qquad \textbf{(B)}\ \frac {23}{128}\qquad \textbf{(C)}\ \frac {1}{4}\qquad \textbf{(D)}\ \frac {35}{128}\qquad \textbf{(E)}\ \frac {1}{2}$

For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "[i]$(P, Q)$-travel route[/i]" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions. [list] [*] $T_0=P,\ T_s=Q$ [*] $T_0, T_1, \ldots, T_s$ are all distinct. [*] There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$. [/list] Prove that there exist two airports $P, Q$ such that there is no or exactly one [i]$(P, Q)$-travel route[/i]. [hide=Graph Wording]Consider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$. [/hide]

Solve the equation $$(12x-1)(6x-1)(4x-1)(3x -1) = 5.$$

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

Triangle $ABC$ has incircle centered at $I.$ Define $M$ and $N$ the midpoints of $BC$ and $CA,$ respectively. Extend $BI$ and $MN$ to meet at a point $K.$ The circumcircle of $\triangle BKC$ intersects the incircle at two points $D$ and $G,$ where $D$ is closer to $AB$ than $G.$ Line $BK$ intersects the incircle at two points $E$ and $F,$ where $FK<EK.$ Let $H$ be $DC \cap BK$. Given that $BD=3$ and $DF=4,$ compute $\tfrac{BE}{EF} \cdot \tfrac{BH}{HF}.$ [i]Tiebreaker #3[/i]

The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0$, and none of $m$, $n$, and $p$ is zero. What is the value of $\frac{n}{p}$? $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 4 \qquad \text{(D)} \ 8\qquad \text{(E)} \ 16$

From point $D$ of parallelogram $ABCD$ were drawn an arbitrary line $\ell_1$, intersecting the segment $AB$ and the line $BC$ at points $C_1$ and $A_1$, respectively, and an arbitrary line $\ell_2$ intersecting the segment $BC$ and the line $AB$ at the points $A_2$ and $C_2$, respectively. Find the locus of the intersection points of the circles $(A_1BC_2)$ and $(A_2BC_1)$ (other than point $B$).

In triangle $ABC$ the incenter is $I$. The center of the excircle opposite $A$ is $I_A$, and it is tangent to $BC$ at $D$. The midpoint of arc $BAC$ is $N$, and $NI$ intersects $(ABC)$ again at $T$. The center of $(AID)$ is $K$. Prove that $TI_A\perp KI$.

Find four different three-digit decimal numbers starting with the same digit, such that their sum is divisible by three of them.

Factor completely the expression $(a-b)^3+(b-c)^3+(c-a)^3$

Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.