Interesting Problems from the AMATYC Student Math League Exams 2011

Interesting problems from the AMATYC Student Math League Exams 2011

(February 2011, #1) After Ed eats 20% of a pie and Ahn eats 40% of a pie, Ed has twice as much left as Ahn. Find Ed’s original amount of pie as a percentage of Ahn’s original amount.

Let E be Ed’s original amount of pie, and A be Ahn’s original amount of pie.

.

So the correct answer is D) 150.

(February 2011, #2) The expression for integers . If , find .

.

The factor pairs of 25 are 1, 25 and 5, 5. This means that and .

So the correct answer is B) 7.

(February 2011, #3) Alicia always climbs steps 1, 2, or 4 at a time. For example, she climbs 4 steps by 1-1-1-1, 1-1-2, 1-2-1,2-1-1,2-2, or 4. In how many ways can she climb 10 steps?

Only 1’s

Ten 1’s

Only 2’s

Five 2’s

1’s and 2’s:

eight 1’s and one 2 / six 1’s and two 2’s / Four 1’s and three 2’s / Two 1’s and four 2’s

1’s and 4’s:

six 1’s and one 4 / two 1’s and two 4’s

2’s and 4’s:

three 2’s and one 4 / one 2 and two 4’s

1’s, 2’s, and 4’s:

four 1’s, one 2, and one 4 / two 1’s, two 2’s, and one 4

This gives .

OR

The number of different ways to get to the 5th step is equal to the number of different ways to get to the 4th step plus the number of different ways to get to the 3rd step plus the number of different ways to get to the 1st step. This is so because the 1st step plus 4 steps gets you to the 5th step, the 3rd step plus 2 steps gets you to the 5th step, and the 4th step plus 1 step gets you to the 5th step. In general, .

With the starting values of , let’s try this new scheme to get the rest. , ,

,

So the correct answer is E) 169. [See the section on Sets and Counting]

(February 2011, #4) The sum of six consecutive positive integers beginning at n is a perfect cube. The smallest such n is 2. Find the sum of the next two smallest such n’s.

.

Since this must be an odd number, we’ll only consider odd cubic numbers:

27, 125, 343, …. 27 gives the value 2, . 729 gives the value 119.

From the list of answer choices, we can check to see if they generate a cubic number. .

So the correct answer is A) 679.

(February 2011, #5) The sum of the infinite geometric series S is 6, and the sum of the series whose terms are the squares of the terms of S is 15. Find the sum of the infinite geometric series with the same first term and opposite common ratio as S.

.

So the correct answer is B) 2.5. [See the section on Algebraic Formulas]

(February 2011, #11) Multiplying the corresponding terms of a geometric and an arithmetic sequence yields 96, 180, 324, 567, …. Find the next term of the new sequence.

, so we get

. Assuming that a and b are whole numbers, , let’s try and . This leads to the system , which has as a solution and . These values lead to the sequences and the product sequence .

So the correct answer is B) 972. [See the section on Algebraic Formulas]

(February 2011, #12) If and , find.

Assuming that x and y are whole numbers, then since , the possible values for x and y are , , , . For the pair , you get .

So the correct answer is B) 36.

(February 2011, #13) The equation (a, b, c positive integers) has a solution in which two of the three numbers are prime. Find the value of the nonprime number.

The possible values of a are 1, 2, 3, and 4. For , we get . For , we get . For , we get . For , we get . We can eliminate and . So now we need to check and .

For , , which is not a square, , which is not a square, , which is not a square, , which is not a square, .

For , , which is , but 18 is not a prime. , which is not a square, , which is , so we get 3, 2, and 42.

So the correct answer is C) 42.

(February 2011, #14) A palindrome is a number like 121 or 1551 which reads the same from right to left and from left to right. How many 4-digit palindromes are divisible by 17?

4-digit palindromes are of the form abba, where a is 1,2,3,4,5,6,7,8,or 9 and b is 0,1,2,3,4,5,6,7,8,9. Now , so we can just examine the 4-digit numbers which are multiples of both 17 and 11, and hence just multiples of 187. We can skip multiples of 10.

/ 1122 / / 3553 / / 5797 / / 7854
/ 1309 / / 3927 / / 5984 / / 8041
/ 1496 / / 4114 / / 6171 / / 8228
/ 1683 / / 4301 / / 6358 / / 8415
/ 2057 / / 4488 / / 6545 / / 8602
/ 2244 / / 4675 / / 6732 / / 8789
/ 2431 / / 4862 / / 6919 / / 8976
/ 2618 / / 5049 / / 7106 / / 9163
/ 2805 / / 5236 / / 7293 / / 9537
/ 3179 / / 5423 / / 7667 / / 9724

And .

So the correct answer is B) 4.

(February 2011, #16) The increasing sequence of positive integers satisfies the equation for all . If , find.

The sequence is . We know that , which means that must be a multiple of 5. If we go with , then , but it doesn’t work. If we go with , then , it works. .

So the correct answer is B) 258.

(October 2011, #1) If the standard order of operations is reversed (that is, additions and subtractions are done first and exponentiation is done last), what is the value of ?

So the correct answer is E) 7776.

(October 2011, #2) The price of a stock rose 20% on Monday, fell 10% on Tuesday, and increased by on Wednesday. By what percent did the price rise from Monday to Wednesday?

So the correct answer is B) 26.

(October 2011, #3) The system of equations and has the solution . Find .

Plugging in leads to. Adding the two equations leads to , so we also get that .

So the correct answer is D) 9.

(October 2011, #4) The positive integersa, b, and c satisfy . Find .

The possible values of a are 1, 2, and 3. These lead to with , with , and with

With , you get

b /
1 / 35.79106
2 / 35.74913
3 / 35.67913
4 / 35.58089
5 / 35.4542
6 / 35.29873
7 / 35.1141
8 / 34.89986
9 / 34.65545
10 / 34.38023
11 / 34.07345
12 / 33.73426
13 / 33.36165
14 / 32.95451
15 / 32.51154
16 / 32.03123
17 / 31.5119
18 / 30.95158
19 / 30.34798
20 / 29.69848
21 / 29

So the correct answer is D) 53.

The previous table can be generated on a TI-83 calculator by pressing the ‘Y =’ key, entering , pressing the ‘TABLE’ key, and scrolling down.

(October 2011, #5) Different shades of pink, red, and white can be made by mixing whole numbers of quarts of red and white paint. Shades are different if the ration of red to white paint is different. Find the number of different possible shades that can be made from at most 4 quarts of red and 5 quarts of white paint.

The total number of ratios is , but some of them are equivalent. 1:1, 2:2, 3:3, and 4:4 are equivalent. 1:2 and 2:4 are equivalent. 2:1 and 4:2 are equivalent. This gets us down to only 15 different ratios.

So the correct answer is A) 15. [See the section on Sets and Counting]

(October 2011, #6) The function has zeros and 6. Find the zeros of .

and

So the correct answer is A) .

(October 2011, #7) One population grows exponentially at the same rate that another population decays exponentially. If the populations were both equal to P on Jan. 1 2009, how will the populations be related on Jan. 1 2012?

and , so .

So the correct answer is B) .

(October 2011, #8) For , both and factor over the integers. Find .

It must be that and

So the correct answer is C) 3.

(October 2011, #9) Ed drives from San Mateo to Atascadero, a distance of 197.5 mi. He starts driving at a constant speed and reduces his speed by 5 mph after each half hour of driving. If the trip takes 3 hr 20 min, how far did he travel in the first 2 hours?

So the correct answer is B) 132.

(October 2011, #10) Sun fills her 10 liter radiator with 20% antifreeze and 80% water. She removes some of the mixture and replaces it with antifreeze. If the radiator is now one quarter antifreeze, how many liters of the original mixture did she remove?

So the correct answer is D) .625.

(October 2011, #11) How many numbers with no more than six digits can be formed using only the digits 1 through 7, with no digit used more than once in a given number?

One-digit numbers: 7

Two-digit numbers:

Three-digit numbers:

Four-digit numbers:

Five-digit numbers:

Six-digit numbers:

So the correct answer is E) 8659. [See the section on Sets and Counting]

(October 2011, #12) The lines with equations and are symmetric with respect to a line with equation with . Find .

The line that we want must bisect the obtuse angle between the two given lines. The acute angle between the lines can be determined from the formula , so the obtuse angle would be . We need half of this angle for the bisector, and this would be . And we need to add this angle to , giving us . The slope of the line we want is the tangent of this angle , . From some trig identities, we get . The line passes through the point of intersection of the given lines which is , so an equation for the line is .

So the correct answer is D) 19. [See the section on Trigonometric Formulas]

OR

Lines with reciprocal slopes are bisected by lines with slope of 1 and :

The lines with equations and have reciprocal slopes, so the bisecting line with positive slope must have slope of 1. [See the section on Equations of Lines]

(October 2011, #13) A square of area 45 is inscribed in a circle C. Find the area of a square inscribed in a semicircle of circle C. (Inscribed means having all 4 vertices on a given figure.)

From November 2003, #13: Square ABCD is inscribed in circle O, and its area is a. Square EFGH is inscribed in a semicircle of O. What is the area of square EFGH?

The area of an inscribed square in a circle of radius r, is .

The area of an inscribed square in a semicircle of radius r, is .

So if , then the area of the square inscribed inside the semicircle is .

Using this, we get that the area of the square inscribed inside the semicircle is .

So the correct answer is B) 18.

(October 2011, #14) The left edge of a dollar bill is folded against the bottom edge to form an isosceles right triangle at the left edge. The new left edge is again folded against the bottom edge. A vertex of the new triangle is the upper right corner of the bill. If a dollar bill is 157 mm long, find its width to the nearest millimeter.

From the labeled diagram, it must be that . This simplifies into . So .

So the correct answer is C) 65.

(October 2011, #15) Five boxes are placed inside an empty box. Each of the 5 new boxes is either left empty or has 5 new boxes placed inside it. This process is repeated until there are 18 boxes containing other boxes. Find the number of empty boxes.

Here are 18 boxes containing other boxes, and there are 73 empty boxes.

So the correct answer is A) 73.

(October 2011, #16) Al, Bo, Cy, and Di are to receive math, physics, chem, and bio awards. Al thinks Di will win bio, Bo thinks Cy will win chem., Cy thinks Al won’t win mth, and Di thinks Bo will win physics. The math and bio winners are both right, and the other winners are both wrong. Who wins the math award?

Let’s try to make 2 of them right and 2 of them wrong:

Al and Bo are right, Cy and Di are wrong.

Al / Bo / Cy / Di
chem / bio

Not possible.

Al and Cy are right, Bo and Di are wrong.

Al / Bo / Cy / Di
chem / math / phys / bio
Al / Bo / Cy / Di
phys / chem / math / bio

Possible, but the math and bio winners aren’t both right.

Al and Di are right, Bo and Cy are wrong.

Al / Bo / Cy / Di
phys / bio

Not possible.

Bo and Cy are right, Al and Di are wrong.

Al / Bo / Cy / Di
bio / math / chem / phys
Al / Bo / Cy / Di
phys / bio / chem / math

Possible, but the math and bio winners aren’t both right.

Bo and Di are right, Al and Cy are wrong.

Al / Bo / Cy / Di
bio / phys / chem / math

Possible, but the math and bio winners aren’t both right.

Cy and Di are right, Al and Bo are wrong.

Al / Bo / Cy / Di
chem / phys / bio / math

This is it.

So the correct answer is D) Di.

(October 2011, #17) The digits 1 through 9 are separated into 3 groups of three digits, and the product of each group is found. Let P be the largest of the 3 products. Find the smallest possible value of P.

74 is not the product of 3 digits 1 through 9. 73 is not the product of 3 digits 1 through 9. 71 is not the product of 3 digits 1 through 9. So it’s down to either 70 or 72. 70 doesn’t work:

, and the other groups would consist of 9, ?, ? and 8, ? , ?, but no matter how you position the 1, 3, 4, and 6, you will get a product that’s larger than 70. With 72, you can get the groups 1, 8, 9 and 6, 3, 4 and 7, 2, 5 and the largest product is 72.

So the correct answer is C) 72.

(October 2011, #18) Out of 10 red chips and 15 green chips, 6 are placed into a bag, 10 into a box, and 9 into a bowl. In how many ways can the chips be distributed, if only the number of red and green chips in each container matters?

Let be the number of red chips in the bag, be the number of red chips in the box, be the number of red chips in the bowl. Let be the number of green chips in the bag, be the number of green chips in the box, be the number of green chips in the bowl. This leads to the system of equations

Which has solutions given by , where s and t are nonnegative integers with . The number of different values of red chips and green chips in each container is equal to the number of different pairs of values of s and t.

This is equal to .

So the correct answer is D) 55.

(October 2011, #19) Square ABCD has side length 72. Let E be the midpoint of side AB , and let and intersect at G. Find the length of the altitude to in .

The area of the square can be calculated as .

This leads to the equation .

This simplifies into .

So the correct answer is D) 24.

(October 2011, #20) Let r be the positive real zero of . The sum can be expressed as the rational number in lowest terms. Find .

Recall the geometric series formula:

.

Notice that

But r is the real zero of , so . Plugging this into the previous equation leads to

.

So the correct answer is E) 130. [See the section on Algebraic Formulas]