Found problems: 60
Compute the remainder when $2^{30}$ is divided by $1000$.
Let
$$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$
Evaluate
$$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$
Consider triangle $\vartriangle ABC$ with $\angle C = 90^o$. Let $P$ be the midpoint of $\overline{AC}$ so that $AP = PC = 1$, and suppose $\angle BAC = \angle CBP$. Compute $AB^2$.
Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.
Let $\triangle ABC$ be a right triangle with hypotenuse $AC$. A square is inscribed in the triangle such that points $D,E$ are on $AC$, $F$ is on $BC$, and $G$ is on $AB$. Given that $AG=2$ and $CF=5$, what is the area of $\triangle BFG$?
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.
Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.
Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.
Nine identical spheres of radius $r$ are packed into a unit cube. One sphere is centered at the center of the cube and is tangent to the other eight spheres, each of which is located in a corner of the cube and is tangent to three faces of the cube. Compute the radius of the spheres $r$.
Suppose that $x,y,z$ are positive real numbers that satisfy
\begin{align*}
x+y+z&=xyz\\
\frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}.
\end{align*}
Compute
$$\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.$$
In the following system of equations
$$|x+y|+|y|=|x-1|+|y-1|=2,$$
find the sum of all possible $x$.
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)
In a given acute triangle $\triangle ABC$ with the values of angles given (known as $a$, $b$, and $c$), the inscribed circle has points of tangency $D,E,F$ where $D$ is on $BC$, $E$ is on $AB$, and $F$ is on $AC$. Circle $\gamma$ has diameter $BC$, and intersects $\overline{EF}$ at points $X$ and $Y$. Find $\tfrac{XY}{BC}$ in terms of the angles $a$, $b$, and $c$.
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?
If $f$ is a permutation of $S = \{0, 1,..., 14\}$, then for integers $k \ge 1$, define $$f^k(x) =\underbrace{f(f...(f(x))... ))}_{k\,\,\, applications \,\,\, of \,\,\, f}$$ Compute the number of permutations $f$ of $S$ such that, for some $k \ge 1$, $f^k(x) = (x + 5) \mod \,\,\, 15$ for all $x \in S$.
Let $ABC$ be a triangle with $AB=2$, $BC=3$, and $AC=4$. Consider all lines $XY$ such that $X$ lies on $AC$, $Y$ lies on $BC$, and $\triangle XYC$ has area equal to half that of $\triangle ABC$. What is the minimum possible length of $XY$?
Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.
$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$.
Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.
For nonzero integers $n$, let $f(n)$ be the sum of all positive integers $b$ for which all solutions $x$ to $x^2 +bx+n = 0$ are integers, and let $g(n)$ be the sum of all positive integers $c$ for which all solutions $x$ to $cx + n = 0$ are integers. Compute $\sum^{2020}_{n=1} (f(n) - g(n))$.
Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value?
There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.
Three unit circles are inscribed inside an equilateral triangle such that each circle is tangent to each of the other $2$ circles and to $2$ sides of the triangle. Compute the area of the triangle.