The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square. [asy] size(130); real r = sqrt(2)/2; defaultpen(linewidth(0.8)); draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r)); [/asy]

The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter

How many ordered sequences of $36$ digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from $0$ to $9$.)

Let $ANC$, $CLB$ and $BKA$ be triangles erected on the outside of the triangle $ABC$ such that $\angle NAC = \angle KBA = \angle LCB$ and $\angle NCA = \angle KAB = \angle LBC$. Let $D$, $E$, $G$ and $H$ be the midpoints of $AB$, $LK$, $CA$ and $NA$ respectively. Prove that $DEGH$ is a parallelogram.

(a) In this drawing, there are three squares on each side of the square. Place a natural number in each of the boxes so that the sum of the numbers of two adjacent boxes is always odd. [img]https://cdn.artofproblemsolving.com/attachments/e/6/75517b7d49857abd3f8f0430a70ae5b0eb1554.gif[/img] (b) In this drawing, there are now four squares on each side of the triangle. Justify why a natural number cannot be placed in each box so that the sum of the numbers in two adjacent boxes is always odd. [img] https://cdn.artofproblemsolving.com/attachments/c/8/061895b9c1cdcb132f7d37087873b7de3fb5f3.gif[/img] (c) If you now draw a polygon with$ 51$ sides and on each side you place $50$ boxes, taking care that there is a box at each vertex. Can you place a natural number in each box so that the sum of the numbers in two adjacent boxes is always odd? Why?

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$. Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.

Shaft section of a circular cone with vertex $P$ is an isosceles right triangle. $A$ is a point on the circle of the bottom surface, while $B$ is a point inside the circle, $O$ is the center of the circle. If $AB\perp OB$ at $B$, $OH\perp PB$ at $H$, $PA=4$, $C$ is the midpoint of $PA$, then when the volume of triangular pyramid $O-HPC$ takes its maximum value, the length of $OB$ is $\text{(A)}\frac{\sqrt5}{3}\qquad\text{(B)}\frac{2\sqrt5}{3}\qquad\text{(C)}\frac{\sqrt6}{3}\qquad\text{(D)}\frac{2\sqrt6}{3}\qquad$

[b]p1.[/b] Warren interrogates the $25$ members of his cabinet, each of whom always lies or always tells the truth. He asks them all, “How many of you always lie?” He receives every integer answer from $1$ to $25$ exactly once. Find the actual number of liars in his cabinet. [b]p2.[/b] Abraham thinks of distinct nonzero digits $E$, $M$, and $C$ such that $E +M = \overline{CC}$. Help him evaluate the sum of the two digit numbers $\overline{EC}$ and $\overline{MC}$. (Note that $\overline{CC}$, $\overline{EC}$, and $\overline{MC}$ are read as two-digit numbers.) [b]p3.[/b] Let $\omega$, $\Omega$, $\Gamma$ be concentric circles such that $\Gamma$ is inside $\Omega$ and $\Omega$ is inside $\omega$. Points $A,B,C$ on $\omega$ and $D,E$ on $\Omega$ are chosen such that line $AB$ is tangent to $\Omega$, line $AC$ is tangent to $\Gamma$, and line $DE$ is tangent to $\Gamma$. If $AB = 21$ and $AC = 29$, find $DE$. [b]p4.[/b] Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes. [b]p5.[/b] At Stillwells Ice Cream, customers must choose one type of scoop and two different types of toppings. There are currently $630$ different combinations a customer could order. If another topping is added to the menu, there would be $840$ different combinations. If, instead, another type of scoop were added to the menu, compute the number of different combinations there would be. [b]p6.[/b] Eleanor the ant takes a path from $(0, 0)$ to $(20, 24)$, traveling either one unit right or one unit up each second. She records every lattice point she passes through, including the starting and ending point. If the sum of all the $x$-coordinates she records is $271$, compute the sum of all the $y$-coordinates. (A lattice point is a point with integer coordinates.) [b]p7.[/b] Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them. [b]p8.[/b] Michelle is playing Spot-It with a magical deck of $10$ cards. Each card has $10$ distinct symbols on it, and every pair of cards shares exactly $1$ symbol. Find the minimum number of distinct symbols on all of the cards in total. [b]p9.[/b] Define the function $f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + ...$ for integers $n \ge 2$. Find $$f(2) + f(4) + f(6) + ... .$$ [b]p10.[/b] There are $9$ indistinguishable ants standing on a $3\times 3$ square grid. Each ant is standing on exactly one square. Compute the number of different ways the ants can stand so that no column or row contains more than $3$ ants. [b]p11.[/b] Let $s(N)$ denote the sum of the digits of $N$. Compute the sum of all two-digit positive integers $N$ for which $s(N^2) = s(N)^2$. [b]p12.[/b] Martha has two square sheets of paper, $A$ and $B$. With each sheet, she repeats the following process four times: fold bottom side to top side, fold right side to left side. With sheet $A$, she then makes a cut from the top left corner to the bottom right. With sheet $B$, she makes a cut from the bottom left corner to the top right. Find the total number of pieces of paper yielded from sheets $A$ and sheets $B$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/ff3a459a135562002aa2c95067f3f01441d626.png[/img] [b]p13.[/b] Let $x$ and $y$ be positive integers such that gcd $(x^y, y^x) = 2^{28}$. Find the sum of all possible values of min $(x, y)$. [b]p14.[/b] Convex hexagon $TRUMAN$ has opposite sides parallel. If each side has length $3$ and the area of this hexagon is $5$, compute $$TU \cdot RM \cdot UA \cdot MN \cdot AT \cdot NR.$$ [b]p15.[/b] Let $x$, $y$, and $z$ be positive real numbers satisfying the system $$\begin{cases} x^2 + xy + y^2 = 25\\ y^2 + yz + z^2 = 36 \\ z^2 + zx + x^2 = 49 \end{cases}$$ Compute $x^2 + y^2 + z^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD \equal{} \angle HFD \equal{} 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that: $ FG \equal{} t \equal{} \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a \equal{} \frac {t \sqrt {3}}{2 \plus{} \sqrt {3}}$.

Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$. Consider the sequence of quadrilaterals $A_iB_iC_iD_i$. i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not? ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?

In a $2025 \times 2025$ table, several cells are marked. At each move, Cyril can get to know the number of marked cells in any checkered square inside the initial table, with side less than $2025$. What is the minimal number of moves, which allows to determine the total number of marked cells for sure? (5 marks)

An unordered triple $(a,b,c)$ in one move can be changed to either of the triples: $(a,b,2a+2b-c)$,$(a,2a+2c-b,c)$ or $(2b+2c-a,b,c)$. Can one get from triple $(3,5,14)$ the triple $(9,8,11)$ in finite amount of moves?

A semicircle is inscribed in right triangle $ABC$ with right angle $B$ and has diameter on $AB$, with one end on point $B$. Given that $AB = 15$ and $BC = 8$, determine the radius of the semicircle [i]2015 CCA Math Bonanza Team Round #6[/i]

Given $2n$ natural numbers, so that the average arithmetic of those $2n$ number is $2$. If all the number is not more than $2n$. Prove we can divide those $2n$ numbers into $2$ sets, so that the sum of each set to be the same.

Let $n$ be a positive integer. Suppose that its positive divisors can be partitioned into pairs (i.e. can be split in groups of two) in such a way that the sum of each pair is a prime number. Prove that these prime numbers are distinct and that none of these are a divisor of $n.$

In a ballroom dance class $15$ boys and $15$ girls are lined up in parallel rows so that $15$ couples are formed. It so happens that the difference in height between the boy and the girl in each couple is not more than $10$ cm. Prove that if the boys and the girls were placed in each line in order of decreasing height, then the difference in height in each of the newly formed couples would still be at most $10$ cm. (AG Pechkovskiy, Moscow)

In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is $01$, one may replace this by substring $100$. The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from $0101$, one can win using move (iii) first in the last two digits, resulting in $01100$, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?

Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds: $$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$

Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that \[ 0< (AB+BC+CD+AD)-(AC+BD) < 4. \]