Found problems: 229
Define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ to be $f(x)=(x+1)!-x!$. Find the number of positive integers $x<49$ such that $f(x)$ divides $f(49)$.
[i]Proposed by David Tang[/i]
There are $5$ people who start with $1, 2, 3, 4,$ and $5$ cookies, respectively. Every minute, two different people are chosen uniformly at random. If they have $a$ and $b$ cookies and $a\neq b,$ the person with more cookies eats $|a-b|$ of their own cookies. If $a = b,$ the minute still passes with nothing happening.
Compute the expected number of minutes until all $5$ people have an equal number of cookies.
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
A positive integer $n$ is [i]brgorable[/i] if it is possible to arrange the numbers $1, 1, 2, 2, ..., n, n$ such that between any two $k$'s there are exactly $k$ numbers (for example, $n=2$ is not brgorable, but $n = 3$ is as demonstrated by $3, 1, 2, 1, 3, 2$). How many brgorable numbers are less than 2019?
Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Raina Yang[/i]
Misha is currently taking a Complexity Theory exam, but he seems to have forgotten a lot of the material! In the question, he is asked to fill in the following boxes with $\subseteq$ and $\subsetneq$ to identify the relationship between different complexity classes: $$\mathsf{NL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{NP}\ \fbox{\phantom{tt}}\ \mathsf{PH}\ \fbox{\phantom{tt}}\ \mathsf{PSPACE}\ \fbox{\phantom{tt}}\ \mathsf {EXP}$$ and $$\mathsf{coNL}\ \fbox{\phantom{tt}}\ \mathsf{P}\ \fbox{\phantom{tt}}\ \mathsf{coNP}\ \fbox{\phantom{tt}}\ \mathsf{PH}$$ Luckily, he remembers that $\mathsf{P} \neq \mathsf{EXP}$, $\mathsf{NL} \neq \mathsf{PSPACE}$, $\mathsf{coNL} \neq \mathsf{PSPACE}$, and $\mathsf{NP} \neq \mathsf{coNP}\implies \mathsf{P}\neq \mathsf{NP} \land \mathsf{P}\neq \mathsf{coNP}$.
How many ways are there for him to fill in the boxes so as not to contradict what he remembers?
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.
Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Raina Yang[/i]
Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.
Let $ABCD$ be a convex quadrilateral with area $202, AB = 4,$ and $\angle A = \angle B = 90^\circ$ such that there is exactly one point $E$ on line $CD$ satisfying $\angle AEB = 90^\circ.$ Compute the perimeter of $ABCD.$
Let $P$ be a point inside isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $$\angle{PAD}=\angle{PDA}=90^\circ-\angle{BPC}.$$ If $PA=14, AB=18,$ and $CD=28,$ compute the area of $ABCD.$
Choose a point $(x,y)$ in the square bounded by $(0,0), (0,1), (1,0)$ and $(1,1)$. Your score is the minimal distance from your point to any other team's submitted point. Your answer must be in the form $(0.abcd, 0.efgh)$ where $a, b, c, d, e, f, g, h$ are decimal digits.
Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$
$\quad \bullet \ f(x+p)=f(x)\text{ and}$
$\quad \bullet \ p \text{ divides } f(x+f(x))-x.$
Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.
A positive integer $n$ is said to be base-able if there exists positive integers $a$ and $b,$ with $b>1,$ such that $n=a^b.$ How many positive integer divisors of $729000000$ are base-able?
[i]Proposed by Kyle Lee[/i]
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this?
[i]Proposed by Nathan Xiong[/i]
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Andy Xu[/i]
Consider the $5$ by $5$ by $5$ equilateral triangular grid as shown:
[asy]
size(5cm);
real n = 5;
for (int i = 0; i < n; ++i) {
draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i));
}
for (int i = 0; i < n; ++i) {
draw((n-i,0)--((n-i)/2,(n-i)*0.866));
}
for (int i = 0; i < n; ++i) {
draw((i,0)--((n+i)/2,(n-i)*0.866));
}
[/asy]
Ethan chooses two distinct upward-oriented equilateral triangles bounded by the gridlines. The probability that Ethan chooses two triangles that share exactly one vertex can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Andrew Wen[/i]
Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$.
[i]Proposed by Nathan Xiong[/i]
Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.
How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\]
Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written such that all that's left on the board are the numbers $1, 2, \dots, T$ in order?
Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that
\[
\gcd(m+1,n+1) = 10\gcd(m,n).
\]
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$
Find the sum of all possible values of $k$