2021 GCIME Problems
Contents
Problem 1
Let denote the number of primes less than or equal to
. Suppose
. For some fixed
what is the maximum possible number of solutions
but not exceeding
?
Problem 2
Let denote the number of solutions to the given equation:
What is the value of
?
Problem 3
Let be a cyclic kite. Let
be the inradius of
. Suppose
is a perfect square. What is the smallest value of
?
Problem 4
Define as the harmonic mean of all the divisors of
. Find the positive integer
for which
is the minimum amongst all
.
Problem 5
Let be a real number such that
If the value of
can be expressed as
where
and
are relatively prime positive integers, then what is the remainder when
is divided by
?
Problem 6
Two scales used to measure temperature are degrees Fahrenheit () and degrees Celsius (
) and the two are related by the formula
. When a two-digit integer degree temperature
in Celcius is converted to Fahrenheit and rounded to the nearest integer degree, it turns out the ones and tens digits of the original Celsius temperature
sometimes switch places to give the rounded Fahrenheit equivalent. Find the sum of all two-digit integer values of
for which this happens.
Problem 7
Let denote the units digit of
. Then find the sum of all positive integers
such that
Problem 8
A basketball club decided to label every basketball in the club. After labeling all of the balls, the labeler noticed that exactly half of the balls had the digit
. Find the sum of all possible three-digit integer values of
.