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AMC 12B 2025

Problems, Step-by-Step Hints, and Solutions

1.
The instructions on a 350-gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires 20 grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?
(A)  16\text{(A)}\;16(B)  17\text{(B)}\;17(C)  18\text{(C)}\;18(D)  19\text{(D)}\;19(E)  20\text{(E)}\;20
2.
Jerry wrote down the ones digit of each of the first 20252025 positive squares: 1,4,9,6,5,6,…1,4,9,6,5,6,\ldots. What is the sum of all the numbers Jerry wrote down?
(A)  9025\text{(A)}\;9025(B)  9070\text{(B)}\;9070(C)  9090\text{(C)}\;9090(D)  9115\text{(D)}\;9115(E)  9160\text{(E)}\;9160
3.
What is the value of i(i−1)(i−2)(i−3)i(i-1)(i-2)(i-3), where i=−1i=\sqrt{-1}?
(A)  6-5i\text{(A)}\;\text{6-5i}(B)  -10i\text{(B)}\;\text{-10i}(C)  10i\text{(C)}\;\text{10i}(D)  −10\text{(D)}\;-10(E)  10\text{(E)}\;10
4.
The value of the two-digit number a‾ b‾\underline{a}\ \underline{b} in base seven equals the value of the two-digit number b‾ a‾\underline{b}\ \underline{a} in base nine. What is a+ba+b?
(A)  7\text{(A)}\;7(B)  9\text{(B)}\;9(C)  10\text{(C)}\;10(D)  11\text{(D)}\;11(E)  14\text{(E)}\;14
5.
Positive integers xx and yy satisfy the equation 57x+22y=40057x+22y=400. What is the least possible value of x+yx+y?
(A)  10\text{(A)}\;10(B)  11\text{(B)}\;11(C)  13\text{(C)}\;13(D)  14\text{(D)}\;14(E)  15\text{(E)}\;15
6.
Emmy says to Max, "I ordered 3636 math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was A‾ B‾ B‾ .B‾ A‾\underline{A}\ \underline{B}\ \underline{B}\ . \underline{B}\ \underline{A}, where AA and BB are digits and A≠0A\ne0." After a pause, Max says, "That was a good price." What is A+BA+B?
(A)  7\text{(A)}\;7(B)  8\text{(B)}\;8(C)  11\text{(C)}\;11(D)  14\text{(D)}\;14(E)  15\text{(E)}\;15
7.
What is the value of ∑n=2255log⁡2(1+1n)(log⁡2n)(log⁡2(n+1))?\sum_{n=2}^{255}\frac{\log_2\left(1+\frac1n\right)}{(\log_2 n)(\log_2(n+1))}?
(A)  34\text{(A)}\;\frac34(B)  1−1log⁡2255\text{(B)}\;1-\frac{1}{\log_2 255}(C)  78\text{(C)}\;\frac78(D)  1516\text{(D)}\;\frac{15}{16}(E)  1\text{(E)}\;1
8.
There are integers aa and bb such that the polynomial x3−5x2+ax+bx^3-5x^2+ax+b has 4+54+\sqrt{5} as a root. What is a+ba+b?
(A)  13\text{(A)}\;13(B)  17\text{(B)}\;17(C)  20\text{(C)}\;20(D)  30\text{(D)}\;30(E)  68\text{(E)}\;68

Solution 1

Solution 2

9.
What is the tens digit of 6666^{6^6}?
(A)  1\text{(A)}\;1(B)  3\text{(B)}\;3(C)  5\text{(C)}\;5(D)  7\text{(D)}\;7(E)  9\text{(E)}\;9

Solution 1

Solution 2

Solution 3

10.
The altitude to the hypotenuse of a 3030-6060-90∘90^\circ right triangle is divided into two segments of lengths x<yx<y by the median to the shortest side of the triangle. What is the ratio xx+y\frac{x}{x+y}?
(A)  37\text{(A)}\;\frac{3}{7}(B)  34\text{(B)}\;\frac{\sqrt3}{4}(C)  49\text{(C)}\;\frac{4}{9}(D)  511\text{(D)}\;\frac{5}{11}(E)  4315\text{(E)}\;\frac{4\sqrt3}{15}
11.
Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing 33 blue bands, 33 red bands, and 33 green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?
(A)  29\text{(A)}\;\frac{2}{9}(B)  27\text{(B)}\;\frac{2}{7}(C)  928\text{(C)}\;\frac{9}{28}(D)  13\text{(D)}\;\frac{1}{3}(E)  38\text{(E)}\;\frac{3}{8}
12.
The windshield wiper on the driver's side of a large bus is depicted below. Arm AB‾\overline{AB} pivots back and forth around point AA, sweeping out an arc of 60∘60^\circ, symmetric about the vertical line through AA. The wiper blade CD‾\overline{CD} is attached to BB at its midpoint and stays vertical as the arm moves. The arm is 33 feet long, and the wiper blade is 3.53.5 feet tall. What is the area of the windshield cleaned by the wiper, in square feet, to the nearest hundredth? (Assume that the windshield is a flat vertical surface.)
Windshield wiper with arm AB and vertical blade CD
(A)  9.68\text{(A)}\;9.68(B)  10.14\text{(B)}\;10.14(C)  10.50\text{(C)}\;10.50(D)  11.32\text{(D)}\;11.32(E)  12.00\text{(E)}\;12.00
13.
A circle has been divided into 66 sectors of different sizes. Then 22 of the sectors are painted red, 22 painted green, and 22 painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below. How many different colorings are possible?
A circle divided into six differently sized sectors colored red, green, and blue
(A)  12\text{(A)}\;12(B)  16\text{(B)}\;16(C)  18\text{(C)}\;18(D)  24\text{(D)}\;24(E)  28\text{(E)}\;28
14.
Consider a decreasing sequence of nn positive integers x1>x2>⋯>xnx_1>x_2>\cdots>x_n that satisfies the following two conditions:

• The average (arithmetic mean) of the first 33 terms in the sequence is 20252025.

• For all 4≤k≤n4\le k\le n, the average of the first kk terms in the sequence is 11 less than the average of the first k−1k-1 terms in the sequence.

What is the greatest possible value of nn?
(A)  1013\text{(A)}\;1013(B)  1014\text{(B)}\;1014(C)  1016\text{(C)}\;1016(D)  2016\text{(D)}\;2016(E)  2025\text{(E)}\;2025
15.
A container has a 1×11\times1 square bottom, a 3×33\times3 open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes 3535 minutes to fill the container up to the midline of the trapezoids. How many more minutes will it take to fill the remainder of the container?
Container with a 1 by 1 square bottom and a 3 by 3 open square top
(A)  70\text{(A)}\;70(B)  85\text{(B)}\;85(C)  90\text{(C)}\;90(D)  95\text{(D)}\;95(E)  105\text{(E)}\;105
16.
An analog clock starts at midnight and runs for 20252025 minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?
(A)  0\text{(A)}\;0(B)  2−1\text{(B)}\;\sqrt2-1(C)  2−2\text{(C)}\;2-\sqrt2(D)  22\text{(D)}\;\frac{\sqrt2}{2}(E)  3−2\text{(E)}\;3-\sqrt2
17.
Each of the 99 squares in a 3×33\times3 grid is to be colored red, blue, or yellow in such a way that each red square shares an edge with at least one blue square, each blue square shares an edge with at least one yellow square, and each yellow square shares an edge with at least one red square. Colorings that can be obtained from one another by rotations and/or reflections are to be considered the same. How many different colorings are possible?
(A)  3\text{(A)}\;3(B)  9\text{(B)}\;9(C)  12\text{(C)}\;12(D)  18\text{(D)}\;18(E)  27\text{(E)}\;27
18.
Awnik repeatedly plays a game that has a probability of winning of 13\frac13. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?
(A)  52\text{(A)}\;\frac52(B)  3\text{(B)}\;3(C)  165\text{(C)}\;\frac{16}{5}(D)  72\text{(D)}\;\frac72(E)  154\text{(E)}\;\frac{15}{4}
19.
A rectangular grid of squares has 141141 rows and 9191 columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from 11 through 141×91=12,831141\times91=12{,}831 into the squares. Horace fills the grid horizontally: he puts 11 through 9191 in order from left to right into row 11, puts 9292 through 182182 into row 22 in order from left to right, and continues similarly through row 141141. Vera fills the grid vertically: she puts 11 through 141141 in order from top to bottom into column 11, then 142142 through 282282 into column 22 in order from top to bottom, and continues similarly through column 9191. How many squares get two copies of the same number?
(A)  7\text{(A)}\;7(B)  10\text{(B)}\;10(C)  11\text{(C)}\;11(D)  12\text{(D)}\;12(E)  19\text{(E)}\;19
20.
A frog hops along the number line according to the following rules.
  • It starts at 00.
  • If it is at 00, then it moves to 11 with probability 12\frac12 and it disappears with probability 12\frac12.
  • For n=1,2,n=1,2, or 33, if it is at nn, then it moves to n+1n+1 with probability 14\frac14, it moves to n−1n-1 with probability 14\frac14, and it disappears with probability 12\frac12.
What is the probability that the frog reaches 44?
(A)  1101\text{(A)}\;\frac{1}{101}(B)  1100\text{(B)}\;\frac{1}{100}(C)  199\text{(C)}\;\frac{1}{99}(D)  198\text{(D)}\;\frac{1}{98}(E)  197\text{(E)}\;\frac{1}{97}
21.
Two non-congruent triangles have the same area. Each triangle has sides of length 88 and 99, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?
(A)  20\text{(A)}\;20(B)  22\text{(B)}\;22(C)  24\text{(C)}\;24(D)  26\text{(D)}\;26(E)  28\text{(E)}\;28
22.
What is the greatest possible area of the triangle in the complex plane with vertices 2z2z, (1+i)z(1+i)z, and (1−i)z(1-i)z, where zz is a complex number satisfying ∣4z−2∣=1|4z-2|=1?
(A)  14\text{(A)}\;\frac14(B)  12\text{(B)}\;\frac12(C)  916\text{(C)}\;\frac{9}{16}(D)  34\text{(D)}\;\frac34(E)  1\text{(E)}\;1
23.
Let SS be the set of all integers z>1z>1 such that for all pairs of nonnegative integers (x,y)(x,y) with x<y<zx<y<z, the remainder when 2025x2025x is divided by zz is less than the remainder when 2025y2025y is divided by zz. What is the sum of the elements of SS?
(A)  3041\text{(A)}\;3041(B)  3542\text{(B)}\;3542(C)  3750\text{(C)}\;3750(D)  4044\text{(D)}\;4044(E)  4319\text{(E)}\;4319
24.
How many real numbers satisfy the equation sin⁡(20πx)=log⁡20(x)\sin(20\pi x)=\log_{20}(x)?
(A)  199\text{(A)}\;199(B)  200\text{(B)}\;200(C)  398\text{(C)}\;398(D)  399\text{(D)}\;399(E)  400\text{(E)}\;400
25.
Three concentric circles have radii 11, 22, 33. An equilateral triangle with side length ss has one vertex on each circle. What is s2s^2?
(A)  6\text{(A)}\;6(B)  254\text{(B)}\;\frac{25}{4}(C)  132\text{(C)}\;\frac{13}{2}(D)  274\text{(D)}\;\frac{27}{4}(E)  7\text{(E)}\;7