MATHEMATICS CHALLENGE PROBLEMS
Students applying to a Summer Institutes mathematics course must submit solutions the following
problems. Your solutions will be useful for us to evaluate your application, and these problems give you
a glimpse of what you might encounter in Summer Institutes mathematics courses.
 Many of these problems are difficult and we do not require you to solve all of them in order to
be admitted to the Summer Institutes. We suggest that you submit solutions to five or more
problems; these should be problems you found challenging or for which you are especially
proud of your solutions.
 Since we evaluate applications on a range of criteria, you will be considered for admission
regardless of the number of solutions you submit, and you are welcome to submit partial
answers or any work toward the solution even if you were not able to solve the problem
completely.
 It is important that you show your work and provide explanations for all of your solutions. A
numerical answer without explanation is not useful for our evaluation. We encourage detailed
carefully written solutions.
 Please work on these problems yourself without assistance of any kind. Please do not ask for help
from a family member, teacher, or anyone else, and do not refer to books, the Internet, or other
sources. If you receive any outside assistance for a problem, you must include a description of
the kind of help you received.
 Do not use a calculator; none of these problems require the use of a calculator or computer.
 There is no time limit other than the deadline for submitting the rest of your Summer Institutes
application. Your answers to these problems cannot be submitted after the rest of your
application is submitted.
1. Find all integers 𝑥 and 𝑦 that satisfy 𝑥 2 − 9𝑦 2 = 2017?
2. When multiplied out, the number 2017! = 1 ∙ 2 ∙ 3 ∙∙∙ 2015 ∙ 2016 ∙ 2017 ends in a
string of zeros. How many zeros are at the end of this number?
3. Determine all integer values 𝑛 for which 𝑛2 + 9𝑛 + 14 is a perfect square. Do not
use a computer or calculator, and be sure to explain how you got your answer. In
particular, explain why no other values of 𝑛, other than the ones you found, make
𝑛2 + 9𝑛 + 14 a perfect square.
4. Find integers 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 such that √3 − √5 is a solution to the equation:
𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒 = 0.
5. The curve below is the concatenation of three semicircles with their centers aligned
vertically, and such that each semicircle has twice the radius of the one below it.
Let 𝐴 and 𝐵 be the leftmost and rightmost points on this curve, respectively. If the
straight-line distance from 𝐴 to 𝐵 is 6 2 , what is the length of the curve in terms of 𝜋?
6. Nine squares are arranged to form a rectangle as shown. If the smallest square (which is
not labeled with a letter) has side of length 1, find the area of the rectangle.


7. Determine the digit a such that the 101-digit number N  88

8a99

9 is divisible by
50
50
7. (The first 50 digits of this number are “8”s, the 51st digit is a, and the last 50 digits
are “9”s). Do not use a computer or calculator, and be sure to explain how you got your
answer.
8. The 15-sided figure below is divided into 13 right triangles. Not all right angles are
marked; however, as you move around the outside of the figure in the counterclockwise
direction, the first corner you meet on each triangle is a right angle. If every side of the
15-sided figure has length 1 except the two sides marked below with lengths 𝑥 and 4,
then what is the value of 𝑥? Note that the figure is not drawn to scale.
7𝑛+1
9. For which positive integers n is the fraction 11𝑛+7 not in lowest terms? Do not use a
computer or calculator, and explain your answer.
10. Explain why it is true that for any set of 56 different integers between 1 and 100
(inclusive) there will be two integers in the set that differ by 11. Then show that it is
possible to have 55 integers between 1 and 100 (inclusive) where no pair in the set have
a difference of 11.