[b]p1.[/b] If the pairwise sums of the three numbers $x$, $y$, and $z$ are $22$, $26$, and $28$, what is $x + y + z$? [b]p2.[/b] Suhas draws a quadrilateral with side lengths $7$, $15$, $20$, and $24$ in some order such that the quadrilateral has two opposite right angles. Find the area of the quadrilateral. [b]p3.[/b] Let $(n)*$ denote the sum of the digits of $n$. Find the value of $((((985^{998})*)*)*)*$. [b]p4.[/b] Everyone wants to know Andy's locker combination because there is a golden ticket inside. His locker combination consists of 4 non-zero digits that sum to an even number. Find the number of possible locker combinations that Andy's locker can have. [b]p5.[/b] In triangle $ABC$, $\angle ABC = 3\angle ACB$. If $AB = 4$ and $AC = 5$, compute the length of $BC$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. [i]Proposed by Ariel García[/i]

The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.

[b]5.[/b] In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$.

[i]20 problems for 20 minutes. [/i] [b]p1.[/b] Evaluate $\frac{\sqrt2 \cdot \sqrt6}{\sqrt3}.$ [b]p2.[/b] If $6\%$ of a number is $1218$, what is $18\%$ of that number? [b]p3.[/b] What is the median of $\{42, 9, 8, 4, 5, 1,13666, 3\}$? [b]p4.[/b] Define the operation $\heartsuit$ so that $i\heartsuit u = 5i - 2u$. What is $3\heartsuit 4$? p5. How many $0.2$-inch by $1$-inch by $1$-inch gold bars can fit in a $15$-inch by $12$-inch by $9$-inch box? [b]p6.[/b] A tetrahedron is a triangular pyramid. What is the sum of the number of edges, faces, and vertices of a tetrahedron? [b]p7.[/b] Ron has three blue socks, four white socks, five green socks, and two black socks in a drawer. Ron takes socks out of his drawer blindly and at random. What is the least number of socks that Ron needs to take out to guarantee he will be able to make a pair of matching socks? [b]p8.[/b] One segment with length $6$ and some segments with lengths $10$, $8$, and $2$ form the three letters in the diagram shown below. Compute the sum of the perimeters of the three figures. [img]https://cdn.artofproblemsolving.com/attachments/1/0/9f7d6d42b1d68cd6554d7d5f8dd9f3181054fa.png[/img] [b]p9.[/b] How many integer solutions are there to the inequality $|x - 6| \le 4$? [b]p10.[/b] In a land for bad children, the flavors of ice cream are grass, dirt, earwax, hair, and dust-bunny. The cones are made out of granite, marble, or pumice, and can be topped by hot lava, chalk, or ink. How many ice cream cones can the evil confectioners in this ice-cream land make? (Every ice cream cone consists of one scoop of ice cream, one cone, and one topping.) [b]p11.[/b] Compute the sum of the prime divisors of $245 + 452 + 524$. [b]p12.[/b] In quadrilateral $SEAT$, $SE = 2$, $EA = 3$, $AT = 4$, $\angle EAT = \angle SET = 90^o$. What is the area of the quadrilateral? [b]p13.[/b] What is the angle, in degrees, formed by the hour and minute hands on a clock at $10:30$ AM? [b]p14.[/b] Three numbers are randomly chosen without replacement from the set $\{101, 102, 103,..., 200\}$. What is the probability that these three numbers are the side lengths of a triangle? [b]p15.[/b] John takes a $30$-mile bike ride over hilly terrain, where the road always either goes uphill or downhill, and is never flat. If he bikes a total of $20$ miles uphill, and he bikes at $6$ mph when he goes uphill, and $24$ mph when he goes downhill, what is his average speed, in mph, for the ride? [b]p16.[/b] How many distinct six-letter words (not necessarily in any language known to man) can be formed by rearranging the letters in $EXETER$? (You should include the word EXETER in your count.) [b]p17.[/b] A pie has been cut into eight slices of different sizes. Snow White steals a slice. Then, the seven dwarfs (Sneezy, Sleepy, Dopey, Doc, Happy, Bashful, Grumpy) take slices one by one according to the alphabetical order of their names, but each dwarf can only take a slice next to one that has already been taken. In how many ways can this pie be eaten by these eight persons? [b]p18.[/b] Assume that $n$ is a positive integer such that the remainder of $n$ is $1$ when divided by $3$, is $2$ when divided by $4$, is $3$ when divided by $5$, $...$ , and is $8$ when divided by $10$. What is the smallest possible value of $n$? [b]p19.[/b] Find the sum of all positive four-digit numbers that are perfect squares and that have remainder $1$ when divided by $100$. [b]p20.[/b] A coin of radius $1$ cm is tossed onto a plane surface that has been tiled by equilateral triangles with side length $20\sqrt3$ cm. What is the probability that the coin lands within one of the triangles? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In rectangle $ABCD$, $AB = 7$ and $AC = 25$. What is its area?

We have a $10 \times 10$ table. $T$ is a set of rectangles with vertices from the table and sides parallel to the sides of the table such that no rectangle from the set is a subrectangle of another rectangle from the set. $t$ is the maximum number of elements of $T$. (a) Prove that $t>300$. (b) Prove that $t<600$. [i]Proposed by Mir Omid Haji Mirsadeghi and Kasra Alishahi[/i]

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.

A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy the equation b=a(a-c). Prove that the triangle is isosceles.

Let $\triangle ABC$ a scalene triangle with incenter $I$, circumcenter $O$ and circumcircle $\Gamma$. The incircle of $\triangle ABC$ is tangent to $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. The line $AI$ meet $EF$ and $\Gamma$ at $N$ and $M\neq A$, respectively. $MD$ meet $\Gamma$ at $L\neq M$ and $IL$ meet $EF$ at $K$. The circumference of diameter $MN$ meet $\Gamma$ at $P\neq M$. Prove that $AK, PN$ and $OI$ are concurrent.

Prove that for an arbitrary $\Delta ABC$ the following inequality holds: $\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

A triangle with sides $a,b,c$ is called a good triangle if $a^2,b^2,c^2$ can form a triangle. How many of below triangles are good? (i) $40^{\circ}, 60^{\circ}, 80^{\circ}$ (ii) $10^{\circ}, 10^{\circ}, 160^{\circ}$ (iii) $110^{\circ}, 35^{\circ}, 35^{\circ}$ (iv) $50^{\circ}, 30^{\circ}, 100^{\circ}$ (v) $90^{\circ}, 40^{\circ}, 50^{\circ}$ (vi) $80^{\circ}, 20^{\circ}, 80^{\circ}$ $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

The altitudes of the acute-angled triangle $ABC$ intersect at the point $H$. On the segments $BH$ and $CH$, the points $B_1$ and $C_1$ are marked, respectively, so that $B_1C_1 \parallel BC$. It turned out that the center of the circle $\omega$ circumscribed around the triangle $B_1HC_1$ lies on the line $BC$. Prove that the circle $\Gamma$, which is circumscribed around the triangle $ABC$, is tangent to the circle $\omega$ .

Inside the square ${ABCD}$, the equilateral triangle $\vartriangle ABE$ is constructed. Let ${M}$ be an interior point of the triangle $\vartriangle ABE$ such that $MB=\sqrt{2}$, $MC=\sqrt{6}$, $MD=\sqrt{5}$ and ${ME=\sqrt{3}}$. Find the area of the square ${ABCD}$.

Let $ A_1,A_2,...,A_{3n} $ be $ 3n\ge 3 $ planar points such that $ A_1A_2A_3 $ is an equilateral triangle and $ A_{3k+1} ,A_{3k+2} ,A_{3k+3} $ are the midpoints of the sides of $ A_{3k-2}A_{3k-1}A_{3k} , $ for all $ 1\le k<n. $ Of two different colors, each one of these points are colored, either with one, either with another. [b]a)[/b] Prove that, if $ n\ge 7, $ then some of these points form a monochromatic (only one color) isosceles trapezoid. [b]b)[/b] What about $ n=6? $

The diagram below shows equilateral $\triangle ABC$ with side length $2$. Point $D$ lies on ray $\overrightarrow{BC}$ so that $CD = 4$. Points $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $E$, $F$, and $D$ are collinear, and the area of $\triangle AEF$ is half of the area of $\triangle ABC$. Then $\tfrac{AE}{AF}=\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + 2n$. [asy] import math; size(7cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; pair A,B,C,D,E,F; B=origin; C=(2,0); D=(6,0); A=(1,sqrt(3)); E=(1/3,sqrt(3)/3); F=extension(A,C,E,D); draw(C--A--B--D,linewidth(1.1)); draw(E--D,linewidth(.7)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$F$",F,NE); [/asy]

Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.

Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$

Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Lewis Chen [/i]

The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is: $\textbf{(A)}\ r^2 \qquad \textbf{(B)}\ r^3 \qquad \textbf{(C)}\ 2r^2 \qquad \textbf{(D)}\ 2r^3 \qquad \textbf{(E)}\ \dfrac{1}{2}r^2$

Let $ABCDEF$ be a convex hexagon, $P, Q, R$ be the intersection points of $AB$ and $EF$, $EF$ and $CD$, $CD$ and $AB$. $S, T,UV$ are the intersection points of $BC$ and $DE$, $DE$ and $FA$, $FA$ and $BC$, respectively. Prove that if $$\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},$$ then $$\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.$$

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

Find the locus of the points of intersection of the altitudes of the triangles inscribed in a given circle.

$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label("$"+lbl[i]+"'$", P, Q); label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]

Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.