Two locations A and B are connected by a 5-mile trail which features a lookout C. A group of 15 hikers started at A and walked along the trail to C. Another group of 10 hikers started at B and walked along the trail to C. The total distance travelled to C by all hikers from the group that started in A was equal to the total distance travelled to C by all hikers from the group that started in B. Find the distance (in miles) from A to C along the trail.

Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200 meter side and run in the same direction, Alice running twice as fast as Bob. After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the building?

Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey. How many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?

Consider the sequence of consecutive even numbers starting from 0, arranged in a staggered format, where each row contains one more number than the previous row. The beginning of this arrangement is shown below: $0$ $2\; 4$ $6\;\underline{8}\;10$ $12\: 14\: 16\: 18$ $20\: 22 \: 24 \: 26\: 28 $ The number in the middle of the third row is 8. What is the number in the middle of the 101-st row?

For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$. Let $s(n)$ denote the sum of the first $n$ factorials, i.e. $$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$Find the remainder when $s(2024)$ is divided by $8$

David wanted to calculate the volume of a prism with an equilateral triangular base. He was given the height of the prism $H=15$ and the height of the base $h=6$. He accidentally swapped the values of $H$ and $h$ in his calculations. By what number should he multiply his result to get the correct volume?

Let $a,b,c,d$ be four distinct integers such that: $$\text{min}(a,b)=2$$$$\text{min}(b,c)=0$$$$\text{max}(a,c)=2$$$$\text{max}(c,d)=4$$ Here $\text{min}(a,b)$ and $\text{max}(a,b)$ denote respectively the minimum and the maximum of two numbers $a$ and $b$. Determine the fifth smallest possible value for $a+b+c+d$

Initially, the integer $80$ is written on a blackboard. At each step, the integer $x$ on the blackboard is replaced with an integer chosen uniformly at random among [0,x−1], unless $x=0$ , in which case it is replaced by an integer chosen uniformly at random among [0,2024]. Let $P(a,b)$ be the probability that after $a$ steps, the integer on the board is $b$. Determine $$\lim_{x\to\infty}\frac{P(a,80)}{P(a,2024)}$$(that is, the value that the function $\frac{P(a,80)}{P(a,2024)}$ approaches as $a$ goes to infinity).

Let the function $f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}$ for all real values $x,y$ and $t\not=\pm1$ a) Evaluate $f(2,0,3)$ and $f(0,2,3)$. b) Show that $f(x,y,0)=f(y,x,0)$ for any values of $(x,y)$. c) Show that $f(x,y,t)=f(y,x,t)$ for any values of $(x,y)$ and $t\not=\pm1$. d) Given $$g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}$$for all real values $x,y$ and $s\not=\frac{\pi}{2}+2\pi k$, where $k$ is an integer number, show that $g(x,y,s)=g(y,x,s)$ for any values of $(x,y)$ and $s$ in the domain of $g(x,y,s)$.

a) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $15$? b) How many ways are there to pair up the elements of $\{1,2,\dots,14\}$ into seven pairs so that each pair has sum at least $13$? c) How many ways are there to pair up the elements of $\{1,2,\dots,2024\}$ into $1012$ pairs so that each pair has sum at least $2001$?

Let $ABC$ be a triangle for which the tangent from $A$ to the circumcircle intersects line $BC$ at $D$, and let $O$ be the circumcenter. Construct the line $l$ that passes through $A$ and is perpendicular to $OD$. $l$ intersects $OD$ at $E$ and $BC$ at $F$. Let the circle passing through $ADO$ intersect $BC$ again at $H$. It is given that $AD=AO=1$. a) Find $OE$ b) Suppose for this part only that $FH=\frac{1}{\sqrt{12}}$: determine the area of triangle $OEF$. c) Suppose for this part only that $BC=\sqrt3$: determine the area of triangle $OEF$. d) Suppose that $B$ lies on the angle bisector of $DEF$. Find the area of the triangle $OEF$.

Call a polynomial $f(x)$ excellent if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$. a) Compute the number of excellent polynomials with degree at most 3. b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$. c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.