Practice (TheColoringMethod)

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Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?


What are the coordinates of the reflection of (6,0) across the graph of $y=3x$?

If the area of a circle's inscribed square is 60, what is the area of its circumscribed square?

What are all values of $x$ for which $log_x\sqrt{x+12}>1$?

Determine the units digit of the sum $0!+1!+2!+\cdots+n!+\cdots+20!$?

What is the area of a trapezoid the lengths of whose bases are 10 and 16, and the lengths of whose legs are 8 and 10?

A diagonal of a square intersects a segment that connects one vertex of the square to the midpoint of an opposite side, as shown. If the length of the shorter section of the diagonal is 2, what is the area of the square?


What real value of $x$ satisfies $\sqrt{5x} - \sqrt{2x} = 5-2$?

How many different ways are there to put 10 different balls into 5 different boxes such that no box is empty or contains more than 4 balls.

What is the units digit of $-1\times 2008 + 2 \times 2007 - 3\times 2006 + 4\times 2005 +\cdots-1003\times 1006 + 1004 \times 1005$?

If $-1 < a < b < 0$, then which relationship below holds? $(A)\quad a < a^3 < ab^2 < ab \qquad (B)\quad a < ab^2 < ab < a^3 \qquad (C)\quad a< ab < ab^2 < a^3 \qquad (D) a^3 < ab^2 < a < ab$

Let point $A$ and $B$ represent real numbers $a$ and $b$, respectively. If $A$ and $B$ lay on different sides of the origin $O$, and $|a - b| = 2016$, $AO = 2BO$, what is the value of $a+b$?

Solve $$\left\{ \begin{array}{rcl} 4x & \equiv 14 &\pmod{15}\\ 9x & \equiv 11 &\pmod{20}\\ \end{array}\right.$$


Show that the sum of all the numbers of the form $\frac{1}{mn}$ is not an integer, where $m$ and $n$ are integers, and $1\le m \le n \le 2017$.

If $a+b=\sqrt{5}$, compute the value of $\frac{a^2 - a^2b^2 + b^2 +2ab}{a+ab+b}+ab$.

If $AE=DE=5$, $AB=CD$, $BC=4$, $\angle{E}=60^\circ$, $\angle{A}=\angle{D}=90^\circ$, then what is the area of the pentagon $ABCDE$?


How many solutions does the following system have? $$ \left\{ \begin{array}{ll} \lfloor x \rfloor + 2y &= 1\\ \lfloor y \rfloor + x &=2 \end{array} \right. $$ Where $\lfloor x \rfloor$ and $\lfloor y \rfloor$ denote the largest integers not exceeding $x$ and $y$, respectively.

Let $a=-2+\sqrt{2}$. Compute $$1+\frac{1}{2+\frac{1}{3+a}}$$

Show that $1^{2017}+2^{2017}+\cdots + n^{2017}$ is not divisible by $(n+2)$ for any positive integer $n$.

How many ordered pairs of integers $(x,y)$ are there such that $x^2 + 2xy+3y^2=34$?


If real numbers $a$ and $b$ satisfy $a^2 + b^2=1$, find the minimal value of $a^4 + ab+b^4$.

Distinct real numbers $a$, $b$ and $c$ satisfy $a+\frac{1}{b}=b+\frac{1}{c} = c+\frac{1}{a}=t$. Find the value of $t$.

How many integers $m$ are there for which $5\times 2^m +1$ is a square number?

If real numbers $a$, $b$ and $c$ satisfy $abc=-1$, $a+b+c=4$, $\frac{a}{a^2-3a-1}+\frac{b}{b^2-3b-1}+\frac{c}{c^2-3c-1}=\frac{4}{9}$, what is the value of $a^2+b^2+c^2$?

Let $\triangle{ABC}$ be a Pythagorean triangle. If $\triangle{ABC}$'s circumstance is 30, find its circumcircle's area.