[b]p1.[/b] Calculate $A \cdot B +M \cdot C$, where $A = 1$, $B = 2$, $C = 3$, $M = 13$. [b]p2.[/b] What is the remainder of $\frac{2022\cdot2023}{10}$ ? [b]p3.[/b] Daniel and Bryan are rolling fair $7$-sided dice. If the probability that the sum of the numbers that Daniel and Bryan roll is greater than $11$ can be represented as the fraction $\frac{a}{b}$ where $a$, $b$ are relatively prime positive integers, what is $a + b$? [b]p4.[/b] Billy can swim the breaststroke at $25$ meters per minute, the butterfly at $30$ meters per minute, and the front crawl at $40$ meters per minute. One day, he swam without stopping or slowing down, swimming $1130$ meters. If he swam the butterfly for twice as long as the breaststroke, plus one additional minute, and the front crawl for three times as long as the butterfly, minus eight minutes, for how many minutes did he swim? [b]p5.[/b] Elon Musk is walking around the circumference of Mars trying to find aliens. If the radius of Mars is $3396.2$ km and Elon Musk is $73$ inches tall, the difference in distance traveled between the top of his head and the bottom of his feet in inches can be expressed as $a\pi$ for an integer $a$. Find $a$. ($1$ yard is exactly $0.9144$ meters). [b]p6.[/b] Lukas is picking balls out of his five baskets labeled $1$,$2$,$3$,$4$,$5$. Each basket has $27$ balls, each labeled with the number of its respective basket. What is the least number of times Lukas must take one ball out of a random basket to guarantee that he has chosen at least $5$ balls labeled ”$1$”? If there are no balls in a chosen basket, Lukas will choose another random basket. [b]p7.[/b] Given $35_a = 42_b$, where positive integers $a$, $b$ are bases, find the minimum possible value of the sum $a + b$ in base $10$. [b]p8.[/b] Jason is playing golf. If he misses a shot, he has a $50$ percent chance of slamming his club into the ground. If a club is slammed into the ground, there is an $80$ percent chance that it breaks. Jason has a $40$ percent chance of hitting each shot. Given Jason must successfully hit five shots to win a prize, what is the expected number of clubs Jason will break before he wins a prize? [b]p9.[/b] Circle $O$ with radius $1$ is rolling around the inside of a rectangle with side lengths $5$ and $6$. Given the total area swept out by the circle can be represented as $a + b\pi$ for positive integers $a$, $b$ find $a + b$. [b]p10.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD$. [b]p11.[/b] Raymond is eating huge burgers. He has been trained in the art of burger consumption, so he can eat one every minute. There are $100$ burgers to start with. However, at the end of every $20$ minutes, one of Raymond’s friends comes over and starts making burgers. Raymond starts with $1$ friend. If each of his friends makes $1$ burger every $20$ minutes, after how long in minutes will there be $0$ burgers left for the first time? [b]p12.[/b] Find the number of pairs of positive integers $(a, b)$ and $b\le a \le 2022$ such that $a\cdot lcm(a, b) = b \cdot gcd(a, b)^2$. [b]p13.[/b] Triangle $ABC$ has sides $AB = 6$, $BC = 10$, and $CA = 14$. If a point $D$ is placed on the opposite side of $AC$ from $B$ such that $\vartriangle ADC$ is equilateral, find the length of $BD$. [b]p14.[/b] If the product of all real solutions to the equation $(x-1)(x-2)(x-4)(x-5)(x-7)(x-8) = -x^2+9x-64$ can be written as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, b, d) = 1$ and $c$ is squarefree, compute $a + b + c + d$. [b]p15.[/b] Joe has a calculator with the keys $1, 2, 3, 4, 5, 6, 7, 8, 9,+,-$. However, Joe is blind. If he presses $4$ keys at random, and the expected value of the result can be written as $\frac{x}{11^4}$ , compute the last $3$ digits of $x$ when $x$ divided by $1000$. (If there are consecutive signs, they are interpreted as the sign obtained when multiplying the two signs values together, e.g $3$,$+$,$-$,$-$, $2$ would return $3 + (-(-(2))) = 3 + 2 = 5$. Also, if a sign is pressed last, it is ignored.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.

Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.

Let $ABC$ be a triangle with $AB = 13$, $AC = 14$, and $BC = 15$, and let $\Gamma$ be its incircle with incenter $ I$. Let $D$ and $E$ be the points of tangency between $\Gamma$ and $BC$ and $AC$ respectively, and let $\omega$ be the circle inscribed in $CDIE$. If $Q$ is the intersection point between $\Gamma$ and $\omega$ and $P$ is the intersection point between $CQ$ and $\omega$, compute the length of $P Q$.

Two externally tangent circles $\Omega_1$ and $\Omega_2$ are internally tangent to the circle $\Omega$ at $A$ and $B$, respectively. If the line $AB$ intersects $\Omega_1$ again at $D$ and $C\in\Omega_1\cap\Omega_2$, show that $\angle BCD=90^\circ$. [i]Proposed by V. Rastorguev[/i]

[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? [b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img] [b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction? [b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? [b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) [b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself). [b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? [b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img] [b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? [b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

The diagram below shows a large circle. Six congruent medium-sized circles are each internally tangent to the large circle and tangent to two neighboring medium-sized circles. Three congruent small circles are mutually tangent to one another and are each tangent to two medium-sized circles as shown. The ratio of the area of the large circle to the area of one of the small circles can be written as $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/a/4/fcffd7ee6e8d3da0641525e7a987d13ce05496.png[/img]

Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$ [img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)

Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

Let $ABC$ be an acute triangle, $k$ its circumcirle and $m$ a line such that $m\cap k=\emptyset, m\parallel BC.$ Denote $D$ the intersection of $m$ and ray $AB.$ a) Let $X$ be an inner point of the arc $BC$ not containing $A$ and denote $Y$ the intersection of lines $m,CX.$ Show that $A,D,X,Y$ are concyclic and name this circle $\kappa$. b) Determine relative position of $\kappa$ and $m$ in case when $C,D,X$ are collinear.

In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.

Let $ABC$ be a right-angled triangle with $\angle B = 90^\circ$ and let $BD$ be the altitude from $B$ on to $AC$. Draw $DE \perp AB$ and $DF \perp BC$. Let $P, Q, R$ and $S$ be respectively the incentres of triangle $DF C, DBF, DEB$ and $DAE$. Suppose $S, R, Q$ are collinear. Prove that $P, Q, R, D$ lie on a circle.

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes. $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$

Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter.Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.

On an arbitrary triangle $ ABC$ let $ E$ be a point on the height from $ A$. Prove that $ (AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.

An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

Consider a trapezoid $ABCD$ with bases $AB$ and $CD$ satisfying $| AB | > | CD |$. Let $M$ be the midpoint of $AB$. Let the point $P$ lie inside $ABCD$ such that $| AD | = | PC |$ and $| BC | = | PD |$. Prove that if $| \angle CMD | = 90^o$, then the quadrilaterals $AMPD$ and $BMPC$ have the same area.

One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.