Found problems: 10
2020 Harvest Math Invitational Team Round Problems, HMI Team #8
8. You have been kidnapped by a witch and are stuck in the [i]Terrifying Tower[/i], which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:[list]
[*] For a floor $i$, put 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3$, 12 on floor $i+2$, 8 on floor $i+1$, and 1 on floor $i$, or remove the corresponding number of boxes from each floor if possible.
[*] For a floor $i$, put 1 box on floor $i+4$, put 3 boxes on floor $i+3$, 6 on floor $i+2$, 5 on floor $i+1$, and 3 on floor $i$, or remove the corresponding number of boxes from each floor if possible. [/list]
At the end, suppose the witch wishes to have exactly $n$ boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let $T(n)$ be the number of distinct distributions of these $n$ boxes that you can make. Find $\displaystyle\sum_{n=1}^{15} T(n)$.
[i]Proposed by Monkey_king1[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #6
6. A triple of integers $(a,b,c)$ is said to be $\gamma$[i]-special[/i] if $a\le \gamma(b+c)$, $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$. For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$, Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$-special. How many distinct values does he write down?
[i]Proposed by winnertakeover[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #9
9. A sequence of nonzero complex numbers $a_1, a_2, \dots, a_{2020}$ satisfies $a_3=a_2^2+2a_1a_2$ and
$$\frac{a_{n+2}}{a_{n+1}}-\frac{a_{n+1}}{a_{n}}=a_n+a_{n+1},$$
for all $2018\ge n\ge 2$. Given $a_2-a_{2020}=2025$, how many integers $0\le a_1\le 2020$ are there, such that $a_1+a_2+\cdots+a_{2019}$ is a real number?
[i]Proposed by winnertakeover[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #2
2. Let $A$ be a set of $2020$ distinct real numbers. Call a number [i]scarily epic[/i] if it can be expressed as the product of two (not necessarily distinct) numbers from $A$. What is the minimum possible number of distinct [i]scarily epic[/i] numbers?
[i]Proposed by Monkey_king1[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #4
4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$?
[i]Proposed by Tragic[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #10
10. Let $p=47$ be a prime. Call a function $f$ defined on the integers [i]lit[/i] if $f(x)$ is an integer from 1 to $p$ inclusive and $f(x+p)=f(x)$ for all integers $x$. How many [i]lit[/i] functions $g$ are there such that for all integers $x$, $p$ divides $g(x^2)-g(x)-x^8+x$?
[i]Proposed by Monkey_king1[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #7
7. In triangle $ABC$, let $N$ and $M$ be the midpoints of $AB$ and $AC$, respectively. Point $P$ is chosen on the arc $BC$ not containing $A$ of the circumcircle of $ABC$ such that $BNMP$ is cyclic. Given $BC=28$, $AC=30$ and $AB = 26$, the value of $AP$ may be expressed as $m/\sqrt{n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Compute $m+n$.
[i]Proposed by winnertakeover[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #5
5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by winnertakeover and Monkey_king1[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #3
3. Let $ABC$ be a triangle with $AB=30$, $BC=14$, and $CA=26$. Let $N$ be the center of the equilateral triangle constructed externally on side $AB$. Let $M$ be the center of the square constructed externally on side $BC$. Given that the area of quadrilateral $ACMN$ can be expressed as $a+b\sqrt{c}$ for positive integers $a$, $b$ and $c$ such that $c$ is not divisible by the square of any prime, compute $a+b+c$.
[i]Proposed by winnertakeover[/i]
2020 Harvest Math Invitational Team Round Problems, HMI Team #1
1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i]