Let $r$ be the radius of the inscribed circle of a right triangle $ABC$. Show that $r$ is less than half of either leg and less than one fourth of the hypotenuse.

Three faces of a rectangular prism have diagonal lengths of $7$, $8$, and $9$ inches. How many cubic inches are in the volume of the rectangular prism? $\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504$

A circle that passes through the vertices $ B,C $ of a triangle $ ABC, $ cuts the segments $ AB,AC $ (endpoints excluded) in $ N, $ respectively, $ M. $ Consider the point $ P $ on the segment $ MN $ and $ Q $ on the segment $ BC $ (endpoints excluded on both segments) such that the angles $ \angle BAC,\angle PAQ $ have the same bisector. Show that: [b]a)[/b] $ \frac{PM}{PN} =\frac{QB}{QC} . $ [b]b)[/b] The midpoints of the segments $ BM,CN,PQ $ are collinear.

$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.

In the triangle $ABC$, $|AB|=15,|BC|=12,|AC|=13$. Let the median $AM$ and bisector $BK$ intersect at point $O$, where $M\in BC,K\in AC$. Let $OL\perp AB,L\in AB$. Prove that $\angle OLK=\angle OLM$.

Given a triangle $\vartriangle ABC$,construct squares $BAQP$ and $ACRS$ outside the triangle $ABC$ (with vertices in that listed in counterclockwise order).Show that the line from $A$ perpendicular to $BC$ passes through the midpoint of the segment $QS$.

We are given a circle $K$ with center $S$ and radius $1$ and a square $Q$ with center $M$ and side $2$. Let $XY$ be the hypotenuse of an isosceles right triangle $XY Z$. Describe the locus of points $Z$ as $X$ varies along $K$ and $Y$ varies along the boundary of $Q.$

Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$ Prove that $A,K,L,E$ are concyclic.

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$? $ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4\sqrt5 \qquad \textbf{(D)}\ 4\sqrt7 \qquad \textbf{(E)}\ 12$

Let $ABCD$ be a square such that $A=(0,0)$ and $B=(1,1)$. $P(\frac{2}{7},\frac{1}{4})$ is a point inside the square. An ant starts walking from $P$, touches $3$ sides of the square and comes back to the point $P$. The least possible distance traveled by the ant can be expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers and $a$ not divisible by any square number other than $1$. What is the value of $(a+b)$?

Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$. [i]Author: Christopher Bradley, United Kingdom [/i]

The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.

In acute $\triangle {ABC}$, ${K}$ is on the extention of segment $BC$. $P, Q$ are two points such that $KP \parallel AB, BK=BP$ and $KQ\parallel AC, CK=CQ$. The circumcircle of $\triangle KPQ$ intersects $AK$ again at ${T}$. Prove that: (1) $\angle BTC+\angle APB=\angle CQA$. (2) $AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP$. Proposed by [i]Yijie He[/i] and [i]Yijuan Yao[/i]

In the acute-angled triangle $ABC$, on the lines $BC$, $AC$, $AB$ we consider the points $D$, $E$ and, respectively, $F$, such that $AD\perp AC, BE\perp AB, CF\perp AC$. Let the point $A', B', C'$ be such that $\{A'\}=BC\cap EF, \{B'\}=AC\cap DF, \{C'\}=AB\cap DE$. Prove that the following inequality is true $$\frac{A'F}{A'E} \cdot \frac{B'D}{B'F} \cdot \frac{C'E}{C'D}\geq8$$

A rhombus $\mathcal R$ has short diagonal of length $1$ and long diagonal of length $2023$. Let $\mathcal R'$ be the rotation of $\mathcal R$ by $90^\circ$ about its center. If $\mathcal U$ is the set of all points contained in either $\mathcal R$ or $\mathcal R'$ (or both; this is known as the [i]union[/i] of $\mathcal R$ and $\mathcal R'$) and $\mathcal I$ is the set of all points contained in both $\mathcal R$ and $\mathcal R'$ (this is known as the [i]intersection[/i] of $\mathcal R$ and $\mathcal R'$, compute the ratio of the area of $\mathcal I$ to the area of $\mathcal U$. [i]Proposed by Connor Gordon[/i]

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$, and let $E$ be the midpoint of the diagonal $BD$. Let $I_1, I_2, I_3, I_4$ be the centers of the circles inscribed into triangles $\vartriangle ABE$, $\vartriangle ADE$, $\vartriangle BCE$, $\vartriangle CDE$, in that order. Prove that the circles $AI_1I_2$ and $CI_3I_4$ intersect $\Gamma$ at diametrically opposite points. Remark: For a circle $C$ and points $X, Y \in C$, we say that $X$ and $Y$ are diametrically opposite if $XY$ is a diameter of $C$.

Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is [asy] pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); transform t = rotate(-45,(3,.5)); pair e = t*a,f=t*b,g=t*c,h=t*d; pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); draw(a--b^^c--d^^e--f^^g--h); filldraw(i--j--l--k--cycle,blue); label("$\alpha$",i+(-.5,.2)); //commented out labeling because it doesn't look right. //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); //draw(lbl1); //label("$1$",lbl1);[/asy] $\text{(A)} \ \sin \alpha \qquad \text{(B)} \ \frac{1}{\sin \alpha} \qquad \text{(C)} \ \frac{1}{1 - \cos \alpha} \qquad \text{(D)} \ \frac{1}{\sin^2 \alpha} \qquad \text{(E)} \ \frac{1}{(1 - \cos \alpha)^2}$

Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.

Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.

Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.

In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$. [i](7 points)[/i]

Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.

Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects $\odot (ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to $\odot(ABC)$ and the tangent at $P$ to $\odot (P QM_a)$ intersect on line $BC$. (Li4)

[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in her set $S$? [b]p2.[/b] Jake has $2021$ balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs? [b]p3.[/b] Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to $(2, 2)$, but she doesn’t know how to get there. So each second, she rides one unit in the positive $x$ or $y$-direction, each with probability $\frac12$ . If the probability that she makes it to $(2, 2)$ during her ride can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, then find $a + b$. [b]p4.[/b] Triangle $ABC$ with $AB = BC = 6$ and $\angle ABC = 120^o$ is rotated about $A$, and suppose that the images of points $B$ and $C$ under this rotation are $B'$ and $C'$, respectively. Suppose that $A$, $B'$ and $C$ are collinear in that order. If the area of triangle $B'CC'$ can be expressed as $a - b\sqrt{c}$ for positive integers $a, b, c$ with csquarefree, find $a + b + c$. [b]p5.[/b] Find the sum of all possible values of $a + b + c + d$ if $a, b, c, $d are positive integers satisfying $$ab + cd = 100,$$ $$ac + bd = 500.$$ [b]p6.[/b] Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points $(1, 2)$ and $(2, 0)$ and a chute-ladder between points $(1, 3)$ and $(4, 0)$, whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from $(0, 0)$ to $(4, 4)$? [b]p7.[/b] There are $8$ identical cubes that each belong to $8$ different people. Each person randomly picks a cube. The probability that exactly $3$ people picked their own cube can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$. Find $a + b$. [b]p8.[/b] Suppose that $p(R) = Rx^2 + 4x$ for all $R$. There exist finitely many integer values of $R$ such that $p(R)$ intersects the graph of $x^3 + 2021x^2 + 2x + 1$ at some point $(j, k)$ for integers $j$ and $k$. Find the sum of all possible values of $R$. [b]p9.[/b] Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$. [b]p10.[/b] In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n \times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [b]p11.[/b] Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2}$$ is an integer. [b]p12.[/b] Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt3$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b}-c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.