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AMC 10B 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.
A Pascal-like triangle has 1010 as the top row and 1010 followed by 11 as the second row. In each subsequent row the first number is 1010, the last number is 11, and, as in the standard Pascal's Triangle, each other number in the row is the sum of the two numbers directly above it. The first four rows are shown below:10101101111021121\begin{array}{ccccccc}&&&10&&&\\ &&10&&1&&\\ &10&&11&&1&\\ 10&&21&&12&&1\end{array}
What is the sum of the digits of the sum of the numbers in the 1111th row?
(A)  11\text{(A)}\;11(B)  13\text{(B)}\;13(C)  14\text{(C)}\;14(D)  16\text{(D)}\;16(E)  17\text{(E)}\;17
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.
In △ABC\triangle ABC, AB=10AB=10, AC=18AC=18, and ∠B=130∘\angle B=130^\circ. Let OO be the center of the circle containing AA, BB, and CC. What is the degree measure of ∠CAO\angle CAO?
(A)  20\text{(A)}\;20(B)  30\text{(B)}\;30(C)  40\text{(C)}\;40(D)  50\text{(D)}\;50(E)  60\text{(E)}\;60
6.
The line y=13x+1y=\frac{1}{3}x+1 divides the square region defined by 0≤x≤20\le x\le2 and 0≤y≤20\le y\le2 into an upper and lower region. The line x=ax=a divides the lower region into two regions of equal area. Then aa can be written as s−t\sqrt{s}-t, where ss and tt are positive integers. What is s+ts+t?
(A)  18\text{(A)}\;18(B)  19\text{(B)}\;19(C)  20\text{(C)}\;20(D)  21\text{(D)}\;21(E)  22\text{(E)}\;22
7.
Frances stands 1515 meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located xx meters east of the locked gate. An unlocked gate lies 99 meters east of the box, and another unlocked gate lies 88 meters west of the locked gate. Frances can reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel is the same via either unlocked gate. What is the value of xx?
(A)  327\text{(A)}\;3\frac{2}{7}(B)  337\text{(B)}\;3\frac{3}{7}(C)  347\text{(C)}\;3\frac{4}{7}(D)  357\text{(D)}\;3\frac{5}{7}(E)  367\text{(E)}\;3\frac{6}{7}
8.
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
9.
How many ordered triples of integers (x,y,z)(x,y,z) satisfy the following system of inequalities?−x−y−z≤−2−x+y+z≤2x−y+z≤2x+y−z≤2\begin{aligned}-x-y-z&\le-2\\ -x+y+z&\le2\\ x-y+z&\le2\\ x+y-z&\le2\end{aligned}
(A)  4\text{(A)}\;4(B)  8\text{(B)}\;8(C)  11\text{(C)}\;11(D)  15\text{(D)}\;15(E)  17\text{(E)}\;17
10.
Let f(n)=n3−5n2+2n+8f(n)=n^3-5n^2+2n+8 and g(n)=n3−6n2+5n+12g(n)=n^3-6n^2+5n+12. What is the sum of all integers nn such that f(n)g(n)\frac{f(n)}{g(n)} is an integer?
(A)  2\text{(A)}\;2(B)  3\text{(B)}\;3(C)  4\text{(C)}\;4(D)  5\text{(D)}\;5(E)  6\text{(E)}\;6
11.
On Monday, 66 students went to the tutoring center at the same time, and each one was randomly assigned to one of the 66 tutors on duty. On Tuesday, the same 66 students showed up, the same 66 tutors were on duty, and the students were again randomly assigned to the tutors. What is the probability that exactly 22 students met with the same tutor both Monday and Tuesday?
(A)  116\text{(A)}\;\frac{1}{16}(B)  316\text{(B)}\;\frac{3}{16}(C)  14\text{(C)}\;\frac{1}{4}(D)  38\text{(D)}\;\frac{3}{8}(E)  12\text{(E)}\;\frac{1}{2}
12.
The figure below shows an equilateral triangle, a rhombus with a 60∘60^\circ angle, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let TT, RR, and HH, respectively, denote the ratio in each case of the total area of the disks to the area of the enclosing polygon. Which of the following is true?
An equilateral triangle, a rhombus, and a regular hexagon containing mutually tangent congruent disks
(A)  T=H=R\text{(A)}\;\text{T=H=R}(B)  H<R=T\text{(B)}\;\text{H<R=T}(C)  H=R<T\text{(C)}\;\text{H=R<T}(D)  H<R<T\text{(D)}\;\text{H<R<T}(E)  H<T<R\text{(E)}\;\text{H<T<R}
13.
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}
14.
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}
15.
The sum ∑k=1∞1k3+6k2+8k\sum_{k=1}^{\infty}\frac{1}{k^3+6k^2+8k} can be expressed as ab\frac{a}{b}, where aa and bb are relatively prime positive integers. What is a+ba+b?
(A)  89\text{(A)}\;89(B)  97\text{(B)}\;97(C)  102\text{(C)}\;102(D)  107\text{(D)}\;107(E)  129\text{(E)}\;129
16.
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
17.
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
18.
What is the ones digit of the sum⌊1⌋+⌊2⌋+⌊3⌋+⋯+⌊2025⌋?\lfloor\sqrt1\rfloor+\lfloor\sqrt2\rfloor+\lfloor\sqrt3\rfloor+\cdots+\lfloor\sqrt{2025}\rfloor?
(Recall that ⌊x⌋\lfloor x\rfloor represents the greatest integer less than or equal to xx.)
(A)  1\text{(A)}\;1(B)  2\text{(B)}\;2(C)  3\text{(C)}\;3(D)  5\text{(D)}\;5(E)  8\text{(E)}\;8
19.
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
20.
Four congruent semicircles are inscribed in a square of side length 11 so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below. The diameter of the small circle can be written as (a+b)(c+d)(\sqrt{a}+b)(\sqrt{c}+d), where a,b,c,a,b,c, and dd are integers. What is a+b+c+da+b+c+d?
Four congruent semicircles inside a square with a small central tangent circle
(A)  3\text{(A)}\;3(B)  5\text{(B)}\;5(C)  8\text{(C)}\;8(D)  9\text{(D)}\;9(E)  11\text{(E)}\;11
21.
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
22.
A seven-digit positive integer is chosen at random. What is the probability that the number is divisible by 1111, given that the sum of its digits is 6161?
(A)  314\text{(A)}\;\frac{3}{14}(B)  311\text{(B)}\;\frac{3}{11}(C)  27\text{(C)}\;\frac{2}{7}(D)  411\text{(D)}\;\frac{4}{11}(E)  37\text{(E)}\;\frac{3}{7}
23.
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
24.
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}
25.
Square ABCDABCD has sides of length 44. Points PP and QQ lie on AD‾\overline{AD} and CD‾\overline{CD}, respectively, with AP=85AP=\frac{8}{5} and DQ=103DQ=\frac{10}{3}. A path begins along the segment from PP to QQ and continues by reflecting against the sides of ABCDABCD (with congruent incoming and outgoing angles). If the path hits a vertex of the square, it terminates there; otherwise it continues forever. At which vertex does the path terminate?
Square ABCD with the path from P to Q
(A)  A\text{(A)}\;\text{A}(B)  B\text{(B)}\;\text{B}(C)  C\text{(C)}\;\text{C}(D)  D\text{(D)}\;\text{D}(E)  The path continues forever.\text{(E)}\;\text{The path continues forever.}