Low Quality Problems
Jeffrey Chen and Kevin Zhao
This is a handout consisting of low quality problems. Calculators are not allowed and
should not be used when completing this problem set. Proceed at your own risk.
Problem 1
Prove Pascal’s Theorem by complex bashing. You must show all your work or else you
will recieve a 0.
Problem 2
Find the last 2 non-zero digits of 2020!. Do not use any theorem learned outside of the
high school curriculum.
Problem 3
Given A = (0, 0), B = (1, 0), and C = (x, y), let the orthocenter and centroid of 4ABC
be H, G respectively. Find the point where the line HG hits the y axis.
Problem 4
How many partitions are there of the set of pairwise distinct triangles with integer side
lengths and a perimeter of 30?
Problem 5
There are 7 trees on a river. Each day, every tree has a 31 chance of turning into a
orange tree. What is the expected number of days for all the trees to turn into orange
trees?
Problem 6
A random subset of the divisors of 420 is chosen. What is the probability that the mean
of the subset is less than or equal to the median of the subset?
Problem 7
Let f (x) = 3x2 +13x+8 and let a, b, c, d, e be real numbers such that a+b+c+d+e = 69.
Find the minimum possible value of f (a) + f (b) + f (c) + f (d) + f (e).
Problem 8
Let 4ABC be a triangle where the incircle with center I hits BC at D and the Aexcircle hits BC at E. If we let the reflection of D over I be G, and let AE = 55, find
(AG + EG)2 .
Problem 9
Let p = 100000007. Let the inverse of a number k be k 0 . How many numbers 0 < x < p
1
Jeffrey Chen and Kevin Zhao
Low Quality Problems
are there such that the inverse of x mod p satisfies the following condition:
|x0 − x| < 42069
Problem 10
What is the probability that randomly chosen ordered pair set S have x > y. (Source:
"Cml Mock")
Problem 11
Find the value of
1
2
+ 16 +
1
12
+
1
20
+
1
30
+
1
42
+
1
.
56
(Source: "Cml Mock")
Problem 12
Find the inradius of a triangle with side lengths 10, 14, and 18. (Source: "November
Contest")
Problem 13
If someone was born in the 1600’s, what is the probability that they will be alive by
the time Newton was born (remember to calculate average life expectancy).
Problem 14
If 1 + 2 + 3 + 4 + . . . + ∞ = − pq where p and q are relatively prime positive integers,
then find p + q.
Problem 15
Find all functions f : R → R such that f (x) + f (y) = f (x + y)
Problem 16
3
Let the silly version of a number n be n! 2 . How many of the first 20 positive integers
have silly counterparts which are integers? (Source: "Mock Mathcounts Series")
Problem 17
What is the 19th digit to the right of the decimal point in the decimal representation
of 712 . (Source: "Mock Mathcounts Series")
2