Test #2 AMATYC Student Mathematics League Winter/Spring 2022

1. A grocery store sells oranges only in small bags of 4 oranges and large bags of 15 oranges (bags cannot be
opened). Kara wants to purchase exactly N oranges, but she cannot since she can only buy whole bags. However,
she would be able to buy any amount greater than N (assuming an unlimited supply of funds and oranges). Find N.
A. 14 B. 27 C. 39 D. 41 E. 43

2. Find the sum of the coefficients of the polynomial obtained by fully expanding (x - 2)5.
A. -2 B. -1 C. 1 D. 2 E. 4

3. A child wishes to color each of the six states on a map as shown. What is the minimum number
of different color crayons she will need to use if each state must be a different color than all states
to which it is adjacent?
A. 2 B. 3 C. 4 D. 5 E. 6

4. On the February 14, 1963 episode of The Twilight Zone entitled “From Agnes –– With Love,” Wally Cox asks
Agnes (the world’s first supercomputer), “What is the smallest prime greater than the 17th root of nine trillion,
three hundred fifty-five million, one hundred twenty-six thousand, six hundred six?” Agnes responds, “Five.”
What is the correct answer?
A. 2 B. 3 C. 5 D. 7 E. 11

𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 = 120
5. Consider the system � . For each solution (x, y, z) with all integer values, compute
𝑥𝑥𝑥𝑥 + 𝑦𝑦𝑦𝑦 − 𝑥𝑥𝑥𝑥 = 52
|𝑥𝑥| + |𝑦𝑦| + |𝑧𝑧|, and then find the total of all of those sums.
A. 32 B. 64 C. 98 D. 128 E. 192

6. The vertices of ∆ABC are labeled with the positive integers a, b, and c (in that order). The midpoint of ����
𝐴𝐴𝐴𝐴 is
���� is labeled 7, and the midpoint of 𝐴𝐴𝐴𝐴
labeled 6, the midpoint of 𝐵𝐵𝐵𝐵 ���� is labeled 5. Suppose the sum of the three
numbers along each side (the two vertices and the midpoint) is the same value for all three sides. Find the smallest
possible value of this common sum.
A. 8 B. 10 C. 12 D. 14 E. 16

7. The number 2022 has three distinct prime factors, a, b, and c. The number 2022! can be factored as
2022! = 𝑎𝑎𝑚𝑚 𝑏𝑏 𝑛𝑛 𝑐𝑐 𝑝𝑝 𝑞𝑞 where q is a natural number not divisible by a, b, or c. Determine m + n + p.
A. 2026 B. 3018 C. 3026 D. 4024 E. 4032
8. The variables a, b, c, d, e, and f are each assigned a distinct value from the set {1, 2, 3, 4, 5, 6} (in other words,
different variables get different values). Find the largest integer less than or equal to the maximum possible value
𝑐𝑐 𝑒𝑒
of the expression 𝑎𝑎𝑎𝑎 + 𝑑𝑑 − 𝑓𝑓 .
A. 30 B. 31 C. 32 D. 33 E. 34

9. Container A holds a 60% acid solution and container B holds an 80% acid solution. The two containers
together hold a total of 300 liters of solutions. A tech removes x liters of solution from each container, then adds
the solution removed from Container A to Container B, and vice versa. After thoroughly mixing the solutions in
each container, container A holds a 64% acid solution and container B holds a 78% acid solution. Find x.
A. 10 B. 20 C. 30 D. 40 E. 50

10. Consider only the points on the graph of 𝑦𝑦 = 𝑥𝑥 2 which have integer coordinates with −10 ≤ 𝑥𝑥 ≤ 10.
Suppose each point with 𝑥𝑥 < 0 is connected to each point with 𝑥𝑥 > 0 via a line segment. What is the sum of all
distinct y-coordinates of the y-intercepts of all such line segments?
A. 1472 B. 1508 C. 1512 D. 1548 E. 1705
11. Three people (X, Y, Z) are in a room with you. One is a knight (knights always tell the truth), one is a knave
(knaves always lie), and the other is a spy (spies may either lie or tell the truth). X says, “I am the knight.” Y says,
“X isn’t lying.” Z says, “I am the spy.” Which of the following correctly identifies all three people?
A. B. C. D. E.
X is the knave. X is the spy. X is the knight. X is the knight X is the knave.
Y is the knight. Y is the knave. Y is the knave. Y is the spy. Y is the spy.
Z is the spy. Z is the knight. Z is the spy. Z is the knave. Z is the knight.

12. If ( f , 0) is the focus of the parabola given by 𝑦𝑦 2 = 12𝑥𝑥, and (h, k) is the center of the hyperbola given by
x2 − y2 − 6x − 8y = 8, find f + h + k.
A. -2 B. -1 C. 0 D. 1 E. 2

13. For how many a ∈ (0, 1) will sin(x) + a sec(x) = cos(x) + a csc(x) have at least two distinct solutions on (0, 2π)?
A. 0 B. 2 C. 4 D. 6 E. Infinitely many

14. The equation 𝑎𝑎4 + 2𝑏𝑏2 + 𝑐𝑐2 = 2022 has six solutions where a, b, c are positive integers. The values of a and c
are both even for all six solutions. Find the only value of b that appears in two different solutions.
A. 1 B. 5 C. 11 D. 19 E. 29

15. A rectangular tabletop is made from 1000 square tiles of the same size with no space between them. It has
dimensions 28" by 17.5". What percentage (rounded to the nearest tenth of a percent) of the total area of the
tabletop consists of edge tiles (that is, tiles that have one or two edges along the outside edge)?
A. 11.5% B. 12.2% C. 12% D. 12.6% E. 13%

16. (i) Is the binary operation a * b = a + b − ab commutative for all real numbers a, b?
(ii) Is the binary operation a * b = a + b − ab associative for all real numbers a, b?
A. (i)Yes (ii)Yes B. (i)Yes (ii)No C. (i)No (ii)Yes D. (i)No (ii)No E. Impossible to determine

17. The position of the hands of a clock are recorded at a moment in time between 4:25:00 and 5:00:00. Call this
time HR:MIN:SEC. 33 minutes and 25 seconds later, the angle measured clockwise from the hour hand to the
minute hand is 10 times greater than it was when the positions were first recorded. To the nearest second, what is
MIN + SEC?
A. 57 B. 61 C. 65 D. 67 E. 77

18. A basketball team plays eight games in a tournament against evenly matched teams (so we may assume that
the probability of the team winning each of the eight games is ½). Find the probability that the team wins at least
three games in a row or loses at least three games in a row at least once during the tournament.
A. 31/64 B. 35/64 C. 39/64 D. 43/64 E. 47/64

19. Nine players are playing Texas Hold ‘Em using a standard deck of 52 cards (2-10, J, Q, K, A in each of 4
suits). Each player is dealt two cards, then five cards are dealt face up for all players to share. Suppose Joe is dealt
two hearts. What is the probability, to the nearest %, that he will make a heart flush (which means that there will
be at least three more hearts dealt face up)?
A. 3% B. 4% C. 5% D. 6% E. 7%

20. Al and Bob are at opposite ends of a diameter of a silo in the shape of a tall right circular cylinder with radius
150 ft. Al is due west of Bob. Al begins walking along the edge of the silo at 6 ft per second at the same moment
that Bob begins to walk due east at the same speed. The value closest to the time in seconds when Al first can see
Bob is:
A. 46 B. 47 C. 48 D. 49 E. 50