[u]Round 1[/u] [b]p1.[/b] Evaluate the sum $1-2+3-...-208+209-210$. [b]p2.[/b] Tony has $14$ beige socks, $15$ blue socks, $6$ brown socks, $8$ blond socks and $7$ black socks. If Tony picks socks out randomly, how many socks does he have to pick in order to guarantee a pair of blue socks? [b]p3.[/b] The price of an item is increased by $25\%$, followed by an additional increase of $20\%$. What is the overall percentage increase? [u]Round 2[/u] [b]p4.[/b] A lamp post is $20$ feet high. How many feet away from the base of the post should a person who is $5$ feet tall stand in order to cast an 8-foot shadow? [b]p5.[/b] How many positive even two-digit integers are there that do not contain the digits $0$, $1$, $2$, $3$ or $4$? [b]p6.[/b] In four years, Jack will be twice as old as Jill. Three years ago, Jack was three times as old as Jill. How old is Jack? [u]Round 3[/u] [b]p7.[/b] Let $x \Delta y = x y^2 -2y$. Compute $20\Delta 18$. [u]p8.[/u] A spider crawls $14$ feet up a wall. If Cheenu is standing $6$ feet from the wall, and is $6$ feet tall, how far must the spider jump to land on his head? [b]p9.[/b] There are fourteen dogs with long nails and twenty dogs with long fur. If there are thirty dogs in total, and three do not have long fur or long nails, how many dogs have both long hair and long nails? [u]Round 4[/u] [b]p10.[/b] Exactly $420$ non-overlapping square tiles, each $1$ inch by $1$ inch, tesselate a rectangle. What is the least possible number of inches in the perimeter of the rectangle? [b]p11.[/b] John drives $100$ miles at fifty miles per hour to see a cat. After he discovers that there was no cat, he drives back at a speed of twenty miles per hour. What was John’s average speed in the round trip? [b]p12.[/b] What percent of the numbers $1,2,3,...,1000$ are divisible by exactly one of the numbers $4$ and $5$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane, such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove that there is a point $P$ in the plane from which all the segments $MN$ are visible at a constant angle.

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1$.Find the largest posible value of $$|x^3+y^3+z^3-xyz|$$

The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E}; draw(A--B--C--cycle); draw(E--D); draw(rightanglemark(E,D,B)); dot(ps); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); label("$3$",midpoint(A--C),NW); label("$4$",midpoint(C--B),NE); label("$5$",midpoint(A--B),SW);[/asy] $ \textbf{(A)}\ \frac{4}{3} \qquad \textbf{(B)}\ \sqrt{5} \qquad \textbf{(C)}\ \frac{9}{4} \qquad \textbf{(D)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(E)}\ \frac{5}{2} $

Let ${ABC}$ be a triangle. Line [i]l[/i] is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on [i]l[/i]. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear. (Slovakia)

In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent. [i]Carl Lian.[/i]

Let $ABC$ be a scalene triangle and $D$ be the feet of altitude from $A$ to $BC$. Let $I_1$, $I_2$ be incenters of triangles $ABD$ and $ACD$ respectively, and let $H_1$, $H_2$ be orthocenters of triangles $ABI_1$ and $ACI_2$ respectively. The circles $(AI_1H_1)$ and $(AI_2H_2)$ meet again at $X$. The lines $AH_1$ and $XI_1$ meet at $Y$, and the lines $AH_2$ and $XI_2$ meet at $Z$. Suppose the external common tangents of circles $(BI_1H_1)$ and $(CI_2H_2)$ meet at $U$. Prove that $UY=UZ$. [i]Proposed by Ivan Chan Kai Chin[/i]

A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively. The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$. Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.

Find the polynomial $f$ with the following properties: $\bullet$ its leading coefficient is $1$, $\bullet$ its coefficients are nonnegative integers, $\bullet$ $72|f(x)$ if $x$ is an integer, $\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.

Determine the smallest possible real constant $C$ such that the inequality $$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$ holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.

Sofía has many boxes where she keeps candies. Every morning, she chooses two of these boxes and places one candy in each. However, each night, a thief selects one box and steals all the candies inside it. Sofía dreams of waking up one day and finding a box with 2024 candies. Prove that Sofía can always fulfill her dream if she has enough boxes.