Olympiad problems

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 #2

Tags: Harvest Math Invitational , HMI , HMMT , set theory

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

Tags: Harvest Math Invitational , HMI , counting , Combo

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 #5

Tags: HMI , Harvest Math Invitational , geometry , euclidean geometry

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

Tags: Harvest Math Invitational , HMI , an epic geometry problem , geometry

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

Tags: Harvest Math Invitational , HMI , HMMT

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]