A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

[b]p1.[/b] Roll two dice. What is the probability that the sum of the rolls is prime? [b]p2. [/b]Compute the sum of the first $20$ squares. [b]p3.[/b] How many integers between $0$ and $999$ are not divisible by $7, 11$, or $13$? [b]p4.[/b] Compute the number of ways to make $50$ cents using only pennies, nickels, dimes, and quarters. [b]p5.[/b] A rectangular prism has side lengths $1, 1$, and $2$. What is the product of the lengths of all of the diagonals? [b]p6.[/b] What is the last digit of $7^{6^{5^{4^{3^{2^1}}}}}$ ? [b]p7.[/b] Given square $ABCD$ with side length $3$, we construct two regular hexagons on sides $AB$ and $CD$ such that the hexagons contain the square. What is the area of the intersection of the two hexagons? [img]https://cdn.artofproblemsolving.com/attachments/f/c/b2b010cdd0a270bc10c6e3bb3f450ba20a03e7.png[/img] [b]p8.[/b] Brooke is driving a car at a steady speed. When she passes a stopped police officer, she begins decelerating at a rate of $10$ miles per hour per minute until she reaches the speed limit of $25$ miles per hour. However, when Brooke passed the police officer, he immediately began accelerating at a rate of $20$ miles per hour per minute until he reaches the rate of $40$ miles per hour. If the police officer catches up to Brooke after 3 minutes, how fast was Brooke driving initially? [b]p9.[/b] Find the ordered pair of positive integers $(x, y)$ such that $144x - 89y = 1$ and $x$ is minimal. [b]p10.[/b] How many zeroes does the product of the positive factors of $10000$ (including $1$ and $10000$) have? [b]p11.[/b] There is a square configuration of desks. It is known that one can rearrange these desks such that it has $7$ fewer rows but $10$ more columns, with $13$ desks remaining. How many desks are there in the square configuration? [b]p12.[/b] Given that there are $168$ primes with $3$ digits or less, how many numbers between $1$ and $1000$ inclusive have a prime number of factors? [b]p13.[/b] In the diagram below, we can place the integers from $1$ to $19$ exactly once such that the sum of the entries in each row, in any direction and of any size, is the same. This is called the magic sum. It is known that such a configuration exists. Compute the magic sum. [img]https://cdn.artofproblemsolving.com/attachments/3/4/7efaa5ba5ad250e24e5ad7ef03addbf76bcfb4.png[/img] [b]p14.[/b] Let $E$ be a random point inside rectangle $ABCD$ with side lengths $AB = 2$ and $BC = 1$. What is the probability that angles $ABE$ and $CDE$ are both obtuse? [b]p15.[/b] Draw all of the diagonals of a regular $13$-gon. Given that no three diagonals meet at points other than the vertices of the $13$-gon, how many intersection points lie strictly inside the $13$-gon? [b]p16.[/b] A box of pencils costs the same as $11$ erasers and $7$ pencils. A box of erasers costs the same as $6$ erasers and a pencil. A box of empty boxes and an eraser costs the same as a pencil. Given that boxes cost a penny and each of the boxes contain an equal number of objects, how much does it costs to buy a box of pencils and a box of erasers combined? [b]p17.[/b] In the following figure, all angles are right angles and all sides have length $1$. Determine the area of the region in the same plane that is at most a distance of $1/2$ away from the perimeter of the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f53ae3b802618703f04f41546e3990a7d0640e.png[/img] [b]p18.[/b] Given that $468751 = 5^8 + 5^7 + 1$ is a product of two primes, find both of them. [b]p19.[/b] Your wardrobe contains two red socks, two green socks, two blue socks, and two yellow socks. It is currently dark right now, but you decide to pair up the socks randomly. What is the probability that none of the pairs are of the same color? [b]p20.[/b] Consider a cylinder with height $20$ and radius $14$. Inside the cylinder, we construct two right cones also with height $20$ and radius $14$, such that the two cones share the two bases of the cylinder respectively. What is the volume ratio of the intersection of the two cones and the union of the two cones? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some unit cubes are stacked atop a flat 4 by 4 square. The figures show views of the stacks from two different sides. Find the maximum and minimum number of cubes that could be in the stacks. Also give top views of a maximum arrangement and a minimum arrangement with each stack marked with its height. [asy] string s = "1010101010111111"; defaultpen(linewidth(0.7)); for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); } }} label("South View",(2,4),N); s = "0101110111111111"; for(int x=0;x<4;++x) { for(int y=0;y<4;++y) { if(hex(substr(s,4*(3-y)+x,1))==1) { x=x+5; draw((x,y)--(x,y+1)--(x+1,y+1)--(x+1,y)--cycle); x=x-5; } }} label("East View",(7,4),N);[/asy]

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

Find the relations among the angles of the triangle $ABC$ whose altitude $AH$ and median $AM$ satisfy $\angle BAH =\angle CAM$.

Let $AB$ be a diameter of a circle $(\omega)$ with centre $O$. From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$. The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$. If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$), then prove that $KC = NL$.

Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$? $\text{(A) }47\qquad\text{(B) }137\qquad\text{(C) }227\qquad\text{(D) }677\qquad\text{(E) }1127$

Equilateral triangle $\vartriangle ABC$ has side length $12$ and equilateral triangles of side lengths $a, b, c < 6$ are each cut from a vertex of $\vartriangle ABC$, leaving behind an equiangular hexagon $A_1A_2B_1B_2C_1C_2$, where $A_1$ lies on $AC$, $A_2$ lies on $AB$, and the rest of the vertices are similarly defined. Let $A_3$ be the midpoint of $A_1A_2$ and define $B_3$, $C_3$ similarly. Let the center of $\vartriangle ABC$ be $O$. Note that $OA_3$, $OB_3$, $OC_3$ split the hexagon into three pentagons. If the sum of the areas of the equilateral triangles cut out is $18\sqrt3$ and the ratio of the areas of the pentagons is $5 : 6 : 7$, what is the value of $abc$?

[u]Round 1[/u] [b]1.1[/b] If the sum of two non-zero integers is $28$, then find the largest possible ratio of these integers. [b]1.2[/b] If Tom rolls a eight-sided die where the numbers $1$ − $8$ are all on a side, let $\frac{m}{n}$ be the probability that the number is a factor of $16$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]1.3[/b] The average score of $35$ second graders on an IQ test was $180$ while the average score of $70$ adults was $90$. What was the total average IQ score of the adults and kids combined? [u]Round 2[/u] [b]2.1[/b] So far this year, Bob has gotten a $95$ and a 98 in Term $1$ and Term $2$. How many different pairs of Term $3$ and Term $4$ grades can Bob get such that he finishes with an average of $97$ for the whole year? Bob can only get integer grades between $0$ and $100$, inclusive. [b]2.2[/b] If a complement of an angle $M$ is one-third the measure of its supplement, then what would be the measure (in degrees) of the third angle of an isosceles triangle in which two of its angles were equal to the measure of angle $M$? [b]2.3[/b] The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that $$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\, \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$ what is the value of $$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$ [u]Round 3[/u] [b]3.1[/b] How many ways can $5$ bunnies be chosen from $7$ male bunnies and $9$ female bunnies if a majority of female bunnies is required? All bunnies are distinct from each other. [b]3.2[/b] If the product of the LCM and GCD of two positive integers is $2021$, what is the product of the two positive integers? [b]3.3[/b] The month of April in ABMC-land is $50$ days long. In this month, on $44\%$ of the days it rained, and on $28\%$ of the days it was sunny. On half of the days it was sunny, it rained as well. The rest of the days were cloudy. How many days were cloudy in April in ABMC-land? [u]Round 4[/u] [b]4.1[/b] In how many ways can $4$ distinct dice be rolled such that a sum of $10$ is produced? [b]4.2[/b] If $p, q, r$ are positive integers such that $p^3\sqrt{q}r^2 = 50$, find the sum of all possible values of $pqr$. [b]4.3[/b] Given that numbers $a, b, c$ satisfy $a + b + c = 0$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}= 10$, and $ab + bc + ac \ne 0$, compute the value of $\frac{-a^2 - b^2 - a^2}{ab + bc + ac}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2826137p24988781]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A_1A_2A_3$ be an acute-angled triangle inscribed into a unit circle centered at $O$. The cevians from $A_i$ passing through $O$ meet the opposite sides at points $B_i$ $(i = 1, 2, 3)$ respectively. [list=a] [*] Find the minimal possible length of the longest of three segments $B_iO$. [*] Find the maximal possible length of the shortest of three segments $B_iO$. [/list]

$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$ [asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((0,0)--(2,1)); draw((0,0)--(1,2)); label("A", (0,0), SW); label("B", (0,2), NW); label("C", (2,2), NE); label("D", (2,0), SE); label("M", (1,2), N); label("N", (2,1), E); label("$\theta$", (.5,.5), SW); [/asy] $ \textbf{(A)}\ \frac{\sqrt{5}}{5} \qquad\textbf{(B)}\ \frac{3}{5} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad\textbf{(D)}\ \frac{4}{5} \qquad\textbf{(E)}\ \text{none of these} $

Point $D$ is the midpoint of $BC$, where $ABC$ is an isosceles triangle ($AB=AC$). On circle $(ABD)$ a point $P \neq A$ is chosen. $O$ is the circumcenter of $ACP$, $Q$ is the foot of the perpendicular from $C$ onto $AO$. Prove that the circumcenter of triangle $ABQ$ lies on the line $AP$

A uniform horizontal circular plate of weight $ Q $ kG is supported at points $ A $, $ B $, $ C $ lying on the circumference of the plate, with $ AC = BC $ and $ ACB = 2\alpha $. What weight $ x $ kG must be placed on the plate at the other end $ D $ of the diameter drawn from point $ C $ so that the pressure of the plate on the support at $ C $G is equal to zero?

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that \[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

Given two points $ A$ and $ B$ and two circles, $\Gamma_1$ with center $A$ and passing through $ B$, and $\Gamma_2$ with center $ B$ and passing through $ A$. Line $AB$ meets $\Gamma_2$ at point $C$. Point $D$ lies on $\Gamma_2$ such that $\angle CDB = 57^o$. Line $BD$ meets $\Gamma_1$ at point $E$. What is $\angle CAE$, in degrees?

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Let $D$ be a point on side $[BC]$ of triangle $ABC$ such that $[AD]$ is an angle bisector, $|BD|=4$, and $|DC|=3$. Let $E$ be a point on side $[AB]$ and different than $A$ such that $m(\widehat{BED})=m(\widehat{DEC})$. If the perpendicular bisector of segment $[AE]$ meets the line $BC$ at $M$, what is $|CM|$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text { None of above} $

Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$. [i]Proposed by Michael Ren[/i]

Given is a triangle $ABC$ with incircle $\omega$, tangent to $BC, CA, AB$ at $D, E, F$. The perpendicular from $B$ to $BC$ meets $EF$ at $M$, and the perpendicular from $C$ to $BC$ meets $EF$ at $N$. Let $DM$ and $DN$ meet $\omega$ at $P$ and $Q$. Prove that $DP=DQ$.

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

On the plane there are two isosceles non-intersecting right triangles $ABC$ and $DEC$ ($AB$ and $DE$ are the hypotenuses,$ ABDE$ is a convex quadrilateral), and $AB = 2 DE$. Let's construct two more isosceles right triangles: $BDF$ (with hypotenuse $BF$ located outside triangle $BDC$) and $AEG$ (with hypotenuse $AG$ located outside triangle $AEC$). Prove that the line $FG$ passes through a point $N$ such that $DCEN$ is a square.

Let $ABC$ be such a triangle, that $AB = 3\cdot BC$. Points $P$ and $Q$ lies on the side $AB$ and $AP = PQ = QB$. A point $M$ is the midpoint of the side $AC$. Prove that $\sphericalangle PMQ = 90^{\circ}$.