Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.
Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 16n = 2016$
Find a polynomial with integral coefficients whose zeros include $\sqrt{2}+\sqrt{5}$.
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.
Let $\triangle{ABC}$ be a right triangle whose sides lengths are all integers. If $\triangle{ABC}$'s perimeter is 30, find its incircle's area.
What is the tens digit of $2015^{2016}-2017?$
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Let even function $f(x)$ and odd function $g(x)$ satisfy the relationship of $f(x)+g(x)=\sqrt{1+x+x^2}$. Find $f(3)$.
Let $f\Big(\dfrac{1}{x}\Big)=\dfrac{1}{x^2+1}$. Compute $$f\Big(\dfrac{1}{2013}\Big)+f\Big(\dfrac{1}{2012}\Big)+f\Big(\dfrac{1}{2011}\Big)+\cdots +f\Big(\dfrac{1}{2}\Big)+f(1)+f(2)+\cdots +f(2011)+f(2012)+f(2013)$$
Let $ABC$ be a right triangle where $AB=4, BC=3$ and $AC=5$. Draw square $ACEF$ and $BCDG$ as shown. Find the area of $\triangle{CDE}$.
Let $ABCD$ be a convex quadrilateral and the two diagonals intersect at point $O$. If $AC = 12, BD=5$ and $\angle{AOB}=2\angle{BOC}$, find the area of $S_{ABCD}$
In $\triangle{ABC}$, if $(b+c):(c+a):(a+b)=5:6:7$, compute $\sin{A}:\sin{B}:\sin{C}$.
In $\triangle{ABC}$, if $\sin{A}:\sin{B}:\sin{C}=4:5:7$, compute $\cos{C}$ and $\sin{C}$.
In $\triangle{ABC}$, if $(a+b+c)(a+b-c)=3ab$, compute $\angle{C}$.
A regular pentagon is inscribed in a unit circle. Find the perimeter of this pentagon.
Let $R=2$ be the circumradius of $\triangle{ABC}$. Compute the value of $$\frac{a+b+c}{\sin{A}+\sin{B}+\sin{C}}$$
In isosceles right triangle $\triangle{ABC}$, $\angle{C}$ is the right angle. Points $D$ and $E$ are on side $AB$ such that $\angle{DCE}=45^\circ$, $AD=25$, and $EB=16$. Let the length of $DE$ be $x$. Find $x^2$.
Let $D$ be a point inside an isosceles triangle $\triangle{ABC}$ where $AB=AC$. If $\angle{ADB} > \angle{ADC}$, prove $DC > DB$.
Let $ABCD$ be a square with side length of 1. Find the total area of the shaded parts in the diagram.
There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
As shown, in quadrilateral $ABCD$, $AB=AD$, $\angle{BAD} = \angle{DCB} = 90^\circ$. Draw altitude from $A$ towards $BC$ and let the foot be $E$. If $AE=1$, find the area of $ABCD$.
In pentagon $ABCDE$, if $AB=AE$, $BC+DE=CD$, and $\angle{ABC} + \angle{AED} = 180^\circ$, show that $\angle{ADE}=\angle{ADC}$.
Let $\triangle{ABC}$ be an isosceles right triangle where $\angle{C}=90^\circ$. If points $M$ and $N$ are on $AB$ such that $\angle{MCN}=45^\circ$, $AM=4$, and $BN=3$, find the length of $MN$.