11/05/2016
RULES
• Work the problems in any order. Some problems are harder
than others; do what you can. If later on you take the
time to figure out how to solve the problems you
could not solve, you will learn more from what you
could not do, than from what you could do.
• If you think you have finished a problem correctly, tell one
of the organizers. If it is really correct, he or she will certify
that it is correct.
• Don’t feel shy about asking for hints.
• Don’t feel shy about getting up, walking around, or talking
with anybody you want to talk to.
• If you want to write on one of the whiteboards, we have
markers available.
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Problems for 11/5/2016
Some problems are again motivated by CRYPTOGRAPHY, secret writing. Modern cryptography uses a lot of
properties of numbers, especially divisibility properties. Some problems are from a past AMC 8 competition.
1. On a certain day in winter The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes,
the sunrise as 6:57 AM, and the sunset as 8:15 PM. The length of daylight and sunrise were correct, but the
sunset was wrong. When did the sun really set?
2. Peter’s family ordered a 12 slice pizza for dinner. Peter ate one slice and shared another slice equally with his
brother. What fraction of the pizza did Peter eat? Reduce the fraction to lowest terms.
3. In the land of Heptakronia the day is divided into seven hours. The picture below shows a typical Heptakronian
clock. Notice that their day begins and ends at 0 hours.
Here Are some questions; hours mean Heptakronian hours. Clock means Heptakronian clock.
(a) If the clock shows 4, what time will it show in 3204 hours?
(b) If it is now 4 o’clock (in Heptakronia), what time was it 7244 hours ago?
(c) It turns out that a Heptakronian hour is divided into 70 minutes, and each minute into 80 seconds
(Heptakronians are weird!) If the time is now 5:37:44, what time will it be in 4 hours, 68 minutes and 40
seconds?
4. Heptakronians love to do what is called arithmetic modulo 7. Here is how it works. If two numbers differ by a
multiple of 7 they are considered not exactly equal, but equal modulo 7. So, for example, Heppi of Heptakronia
will say 6 × 5 = 16 (mod 7), and Heppi is right! The value in regular arithmetic is 30, but 30 − 16 = 14, and 14
is a multiple of 7. But what Heptakronians really like to when practicing arithmetic modulo 7 is to also reduce
the answer so it is a number in the range 0, 1, 2, 3, 4, 5, 6. So Peppi, Heppi’s friend calculates 6 × 5 = 2 (mod
7); much better. As mentioned, two numbers are equal modulo 7 if their difference is a multiple of 7. They are
also equal modulo 7 if the have the same remainder when divided by 7. Can you solve the following equations
mod 7? Your answer in every case should be an integer in the range 0–6.
(a) 5x = 3 (mod 7)
(b) x2 = 2 (mod 7)
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(c) 3x2 − 1 = 0 (mod 7)
(d) x6 = 1 (mod 7)
5. What is the unit digit of 132016 ?
6. Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five digit numbers so that they
have the largest possible sum. Which of the following could be one of the numbers?
(A) 76531
(B) 86724
(C) 87431
(D) 96240
(E) 97403
7. Let ABCD be a quadrilateral that has an inscribed circle. If |AB| = 10, |BC| = 5, and |CD| = 3, what is the
length of AD?
8. A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown.
What is the ratio of the area of the star figure to the area of the original circle? The answer should have the
a + bπ
form
, where a, b are integers, not necessarily positive.
π
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9. Triangle ABC is an equilateral triangle. A point P inside the triangle is situated so that if Q, R, and S are the
feet of the perpendiculars of P to sides AB, BC, and CA respectively, then |P Q| = 1, P R = 2, and |P S| = 3.
What is |AB|?
10. This is a problem that appeared in a college journal. But the solution is based on a simple trick. All you need
to know is that the sum of the angles of a triangle add up to 180◦ .
A triangle is inscribed in a square, as shown in the picture. What is the measure of the angle with the question
mark?