Let $ABC$ be an equilateral triangle, and $P$ be an arbitrary point within the triangle. Perpendiculars $PD,PE,PF$ are drawn to the three sides of the triangle. Show that, no matter where $P$ is chosen, \[ \frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}. \]

Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value $a$.

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

Let $ABC$ be a triangle. The angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at $D$. If $\angle BAC =80^o$ , what are all possible values for $\angle BDC$ ?

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

In a country named Fillip, there are three major cities called Alenda, Breda, Chenida. This country uses the unit of "FP". The distance between Alenda and Chenida is $100$ FP. Breda is $70$ FP from Alenda and $30$ FP from Chenida. Let us say that we take a road trip from Alenda to Chenida. After $2$ hours of driving, we are currently at $50$ FP away from Alenda and $50$ FP away from Chenida. How many FP are we away from Breda?

A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$. $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 18$

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is $11^2 - 9^2$? [b]R2.[/b] Write $\frac{9}{15}$ as a decimal. [b]R3.[/b] A $90^o$ sector of a circle is shaded, as shown below. What percent of the circle is shaded? [b]R4.[/b] A fair coin is flipped twice. What is the probability that the results of the two flips are different? [b]R5.[/b] Wayne Dodson has $55$ pounds of tungsten. If each ounce of tungsten is worth $75$ cents, and there are $16$ ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth? [b]R6.[/b] Tenley Towne has a collection of $28$ sticks. With these $28$ sticks he can build a tower that has $1$ stick in the top row, $2$ in the next row, and so on. Let $n$ be the largest number of rows that Tenley Towne’s tower can have. What is n? [b]R7.[/b] What is the sum of the four smallest primes? [b]R8 / P1.[/b] Let $ABC$ be an isosceles triangle such that $\angle B = 42^o$. What is the sum of all possible degree measures of angle $A$? [b]R9.[/b] Consider a line passing through $(0, 0)$ and $(4, 8)$. This line passes through the point $(2, a)$. What is the value of $a$? [b]R10 / P2.[/b] Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled? [b]R11.[/b] Guang chooses $4$ distinct integers between $0$ and $9$, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number? [b]R12 / P4.[/b] David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex $n$-gon, and it so happens that every angle he assigned is less than $144$ degrees. He tells Pratik the value of $n$ and the degree measures in the $n$-gon, and to David’s dismay, Pratik claims that such an $n$-gon does not exist. What is the smallest value of $n \ge 3$ such that Pratik’s claim is necessarily true? [b]R13 / P3.[/b] Consider a triangle $ABC$ with side lengths of $5$, $5$, and $2\sqrt5$. There exists a triangle with side lengths of $5, 5$, and $x$ ($x \ne 2\sqrt5$) which has the same area as $ABC$. What is the value of $x$? [b]R14 / P5.[/b] A mother has $11$ identical apples and $9$ identical bananas to distribute among her $3$ kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana? [b]R15 / P7.[/b] Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes. [b]P6.[/b] Srinivasa Ramanujan has the polynomial $P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12$. His friend Hardy tells him that $3$ is one of the roots of $P(x)$. What is the sum of the other roots of $P(x)$? [b]P8.[/b] $ABC$ is an equilateral triangle with side length $10$. Let $P$ be a point which lies on ray $\overrightarrow{BC}$ such that $PB = 20$. Compute the ratio $\frac{PA}{PC}$. [b]P9.[/b] Let $ABC$ be a triangle such that $AB = 10$, $BC = 14$, and $AC = 6$. The median $CD$ and angle bisector $CE$ are both drawn to side $AB$. What is the ratio of the area of triangle $CDE$ to the area of triangle $ABC$? [b]P10.[/b] Find all integer values of $x$ between $0$ and $2017$ inclusive, which satisfy $$2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).$$ [b]P11.[/b] Let $x^2 + ax + b$ be a quadratic polynomial with positive integer roots such that $a^2 - 2b = 97$. Compute $a + b$. [b]P12.[/b] Let $S$ be the set $\{2, 3, ... , 14\}$. We assign a distinct number from $S$ to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct) [b]P13.[/b] In triangle $ABC$, $AB = 10$, $BC = 21$, and $AC = 17$. $D$ is the foot of the altitude from $A$ to $BC$, $E$ is the foot of the altitude from $D$ to $AB$, and $F$ is the foot of the altitude from $D$ to $AC$. Find the area of the smallest circle that contains the quadrilateral $AEDF$. [b]P14.[/b] What is the greatest distance between any two points on the graph of $3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11$? [b]P15.[/b] For a positive integer $n$, $\tau (n)$ is defined to be the number of positive divisors of $n$. Given this information, find the largest positive integer $n$ less than $1000$ such that $$\sum_{d|n} \tau (d) = 108.$$ In other words, we take the sum of $\tau (d)$ for every positive divisor $d$ of $n$, which has to be $108$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.

Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.

Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.

What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below? [asy] size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--(4,0)); draw((1,1)--(4,1)); draw((0,2)--(3,2)); draw((0,3)--(2,3)); [/asy] $\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$

In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$. [i]Proposed by Evan Chen[/i]

A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$. Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$. Prove that $\angle BXC=90^{\circ}$.

The point $K{}$ is the middle of the arc $BAC$ of the circumcircle of the triangle $ABC$. The point $I{}$ is the center of its inscribed circle $\omega$. The line $KI$ intersects the circumcircle of the triangle $ABC$ at $T{}$ for the second time. Prove that the circle passing through the midpoints of the segments $BC, BT$ and $CT$ is tangent to the circle which is symmetric to $\omega$ with respect to $BC$.

There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the intersection and one of $A$, $B$ and $C$ of his choice. Find the maximum possible number of three-line intersections within $ABC$ after 300 turns. [i] Proposed by usjl[/i]

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $