Prove that if the length and breadth of a rectangle are both odd integers, then there does not exist a point $P$ inside the rectangle such that each of the distances from $P$ to the 4 corners of the rectangle is an integer.

Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.

In triangle $ABC,$ $AB=BC,$ and let $I$ be the incentre of $\triangle ABC.$ $M$ is the midpoint of segment $BI.$ $P$ lies on segment $AC,$ such that $AP=3PC.$ $H$ lies on line $PI,$ such that $MH\perp PH.$ $Q$ is the midpoint of the arc $AB$ of the circumcircle of $\triangle ABC$. Prove that $BH\perp QH.$

A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] obtuse-angled[/i] set can be extended to an infinite [i]obtuse-angled[/i] set? (UK)

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

Let $ABC$ be a triangle and $P$ a point in its interior. Circle $\Gamma_A$ is considered such that it is tangent to rays $(PB$ and $(PC$. Define similarly $\Gamma_B$ and $\Gamma_C$. Let $\ell_A\neq PA$ be the other common internal tangent of $\Gamma_B$ and $\Gamma_C$. Prove that $\ell_A$, $\ell_B$ and $\ell_C$ meet at a point. [i]Proposed by Andrei Vila[/i]

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that $$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$ (a) Compute $\lambda(2)$. (b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$. (c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. The square $BDEF$ is inscribed in $\triangle ABC$, such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$, respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the square $BDEF$, such that the radius of $\omega_1$ and $\omega_3$ are both equal to $b$ and the radius of $\omega_2$ and $\omega_4$ are both equal to $a$. The circle $\omega_1$ is tangent to $ED$, the circle $\omega_3$ is tangent to $BF$, $\omega_2$ is tangent to $EF$ and $\omega_4$ is tangent to $BD$, each one of these circles are tangent to the two closest circles and the circles $\omega_1$ and $\omega_3$ are tangents. Determine the ratio $\frac{c}{a}$.

Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point

A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$. Prove that $XN=NY$.

Three ants are at three corners of a rectangle. It is assumed that each ant moves only when the other two are stopped and always parallel to the line defined by them. Will be is it possible that the three ants are simultaneously at midpoints on the sides of the rectangle?

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly. (a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$. (b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle. [i]M. Marinov and N. Beluhov[/i]

Let $ ABC$ be an acute triangle. Point $ D$ lies on side $ BC$. Let $ O_B, O_C$ be the circumcenters of triangles $ ABD$ and $ ACD$, respectively. Suppose that the points $ B, C, O_B, O_C$ lies on a circle centered at $ X$. Let $ H$ be the orthocenter of triangle $ ABC$. Prove that $ \angle{DAX} \equal{} \angle{DAH}$. [i]Zuming Feng.[/i]

Let $ABCD$ be a cyclic quadrilateral with $\angle ADC = \angle DBA$. Furthermore, let $E$ be the projection of $A$ on $BD$. Show that $BC = DE - BE$ .

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.

Find the reflection of the point $(11, 16, 22)$ across the plane $3x+4y+5z=7$.

Regular tetrahedron $P_1P_2P_3P_4$ has side length $1$. Define $P_i$ for $i > 4$ to be the centroid of tetrahedron $P_{i-1}P_{i-2}P_{i-3}P_{i-4}$, and $P_{ \infty} = \lim_{n\to \infty} P_n$. What is the length of $P_5P_{ \infty}$?

[b]p1.[/b] $17.5\%$ of what number is $4.5\%$ of $28000$? [b]p2.[/b] Let $x$ and $y$ be two randomly selected real numbers between $-4$ and $4$. The probability that $(x - 1)(y - 1)$ is positive can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p3.[/b] In the $xy$-plane, Mallen is at $(-12, 7)$ and Anthony is at $(3,-14)$. Mallen runs in a straight line towards Anthony, and stops when she has traveled $\frac23$ of the distance to Anthony. What is the sum of the $x$ and $y$ coordinates of the point that Mallen stops at? [b]p4.[/b] What are the last two digits of the sum of the first $2021$ positive integers? [b]p5.[/b] A bag has $19$ blue and $11$ red balls. Druv draws balls from the bag one at a time, without replacement. The probability that the $8$th ball he draws is red can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p6.[/b] How many terms are in the arithmetic sequence $3$, $11$, $...$, $779$? [b]p7.[/b] Ochama has $21$ socks and $4$ drawers. She puts all of the socks into drawers randomly, making sure there is at least $1$ sock in each drawer. If $x$ is the maximum number of socks in a single drawer, what is the difference between the maximum and minimum possible values of $x$? [b]p8.[/b] What is the least positive integer $n$ such that $\sqrt{n + 1} - \sqrt{n} < \frac{1}{20}$? [b]p9.[/b] Triangle $\vartriangle ABC$ is an obtuse triangle such that $\angle ABC > 90^o$, $AB = 10$, $BC = 9$, and the area of $\vartriangle ABC$ is $36$. Compute the length of $AC$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/b648d0d60c186d01493fcb4e21b5260c46606e.png[/img] [b]p10.[/b] If $x + y - xy = 4$, and $x$ and $y$ are integers, compute the sum of all possible values of$ x + y$. [b]p11.[/b] What is the largest number of circles of radius $1$ that can be drawn inside a circle of radius $2$ such that no two circles of radius $1$ overlap? [b]p12.[/b] $22.5\%$ of a positive integer $N$ is a positive integer ending in $7$. Compute the smallest possible value of $N$. [b]p13.[/b] Alice and Bob are comparing their ages. Alice recognizes that in five years, Bob's age will be twice her age. She chuckles, recalling that five years ago, Bob's age was four times her age. How old will Alice be in five years? [b]p14.[/b] Say there is $1$ rabbit on day $1$. After each day, the rabbit population doubles, and then a rabbit dies. How many rabbits are there on day $5$? [b]15.[/b] Ajit draws a picture of a regular $63$-sided polygon, a regular $91$-sided polygon, and a regular $105$-sided polygon. What is the maximum number of lines of symmetry Ajit's picture can have? [b]p16.[/b] Grace, a problem-writer, writes $9$ out of $15$ questions on a test. A tester randomly selects $3$ of the $15$ questions, without replacement, to solve. The probability that all $3$ of the questions were written by Grace can be written in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p17.[/b] Compute the number of anagrams of the letters in $BMMTBMMT$ with no two $M$'s adjacent. [b]p18.[/b] From a $15$ inch by $15$ inch square piece of paper, Ava cuts out a heart such that the heart is a square with two semicircles attached, and the arcs of the semicircles are tangent to the edges of the piece of paper, as shown in the below diagram. The area (in square inches) of the remaining pieces of paper, after the heart is cut out and removed, can be written in the form $a-b\pi$, where $a$ and $b$ are positive integers. Compute $a + b$. [b]p19.[/b] Bayus has $2021$ marbles in a bag. He wants to place them one by one into $9$ different buckets numbered $1$ through $9$. He starts by putting the first marble in bucket $1$, the second marble in bucket $2$, the third marble in bucket $3$, etc. After placing a marble in bucket $9$, he starts back from bucket $1$ again and repeats the process. In which bucket will Bayus place the last marble in the bag? [img]https://cdn.artofproblemsolving.com/attachments/9/8/4c6b1bd07367101233385b3ffebc5e0abba596.png[/img] [b]p20.[/b] What is the remainder when $1^5 + 2^5 + 3^5 +...+ 2021^5$ is divided by $5$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to this circle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line through P parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point. :)

Let $t$ be a positive constant. Given two points $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$ in a space with the origin $O$. Suppose mobile points $P$ in such way that $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of $t$ such that the maximum value of $OP$ is 3.