math

1. A souvenir shop sells solid metal sculptures of Achilles in two sizes. One is 6 cm tall and one is 8 cm tall. Proportions between the two sculptures are preserved. The smaller sculpture has a mass of 486 grams. What is the mass, in grams, of the larger sculpture? (A) 648 (B) 864 (C) 972 (D) 1152 (E) 1458

D. 1152. Mass scales as the cube of the linear scale factor, so 486(8/6)³ = 1152.

2. Bees occupy the 8 corners of a 1 × 2 × 3 rectangular box. The bees can fly inside the box, but cannot fly farther than 1/2 unit from their respective corners. What is the volume, in cubic units, of the region inside the box that cannot be reached by any bee? (A) 6 − π/3 (B) 6 − π/4 (C) 6 − π/6 (D) 6 − π/8 (E) 6 − π/12

C. 6 − π/6. Eight octants with disjoint interiors have the volume of one sphere of radius 1/2. Subtract its volume, π/6, from 6.

3. There are 2026 equally spaced points that all lie on a circle. A line segment is drawn between two points if there is exactly one point between them along the circle. Into how many regions will this divide the interior of the circle? The figure below shows an example with 10 equally spaced points, in which case there are 21 interior regions. (A) 4052 (B) 4053 (C) 4054 (D) 6079 (E) 6081

B. 4053. Each of 2026 chords crosses two others, for 2026 distinct crossings. Each chord adds one region plus one per crossing with an earlier chord. The total is 1 + 2026 + 2026.

4. For how many integers n is (n² − 1)/(n + 2) also an integer? (A) 2 (B) 3 (C) 4 (D) 6 (E) infinitely many

C. 4. Division gives n − 2 + 3/(n + 2). Thus n + 2 can be any of −3, −1, 1, or 3.

5. Let a = log₂ 27, let b = log₃ 25, and let c = log₃₂ 5. What is ab/c? (A) 25 (B) 27 (C) 30 (D) 32 (E) 36

C. 30. The product ab is 6 log₂(3) log₃(5) = 6 log₂(5), and c = log₂(5)/5.

6. The integer n satisfies the inequalities n/100 < 1/(5 · 10) + 1/(10 · 15) + 1/(15 · 20) + ⋯ + 1/(90 · 95) + 1/(95 · 100) < (n + 1)/100. What is n? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

B. 3. Write 1/[k(k + 5)] as (1/5)[1/k − 1/(k + 5)]. The sum telescopes to 19/500 = 0.038.

7. Sharon tosses a fair coin six times and keeps track of how many heads and tails she has gotten. Her first toss is heads. What is the probability that after every toss Sharon’s count of heads differs from her count of tails? (A) 1/8 (B) 3/16 (C) 1/4 (D) 5/16 (E) 3/8

D. 5/16. The second toss must be heads. HHH has seven valid endings. HHT requires a fourth head and then has three valid endings. There are 10 valid outcomes out of 32.

8. A regular octahedron has 8 equilateral triangle faces with edges of length √2. A plane is perpendicular to the line segment joining two opposite vertices of the octahedron and intersects that line segment 1/2 unit from one of its endpoints. What is the area of the intersection of that plane and the octahedron? (A) 1/4 (B) 1/2 (C) (1/2)√2 (D) 3/4 (E) √2

B. 1/2. The equatorial square has area 2 and sits 1 unit from the apex. The halfway slice has half the side length and one-quarter the area.

9. Suppose that a and b are positive integers satisfying 20a + 26b = 2026. What is the least possible value of a + b? (A) 71 (B) 80 (C) 83 (D) 86 (E) 88

B. 80. From 10a + 13b = 1013, we have b ≡ 1 (mod 10). Increasing b lowers a + b. The largest feasible b is 71, giving a = 9.

10. Point C lies on a circle with diameter A̅B̅. Point D lies on A̅B̅, 14 units from A, 24 units from B, and 16 units from C. What is the area, in square units, of △ABC? (A) 186√2 (B) 152√3 (C) 264 (D) 187√2 (E) 108√6

B. 152√3. Put the circle center at (0,0) and D at (−5,0). Subtract x² + y² = 361 from (x + 5)² + y² = 256 to find x = −13. The height is 8√3 and the base is 38.

11. A list contains 2026 statements, each of which is either true or false. The first statement is “The number of false statements in this list is at least 1001.” The second statement is “The number of false statements in this list is at least 1002.” In general, the kth statement in the list is “The number of false statements in this list is at least k + 1000.” How many statements in the list are true? (A) 342 (B) 513 (C) 1013 (D) 1026 (E) 1513

B. 513. If t statements are true, the true statements are exactly those with k ≤ 1026 − t. Solve t = 1026 − t.

12. Polly and Quentin are dancing. They start 1 meter apart. Polly rotates 120° clockwise around Quentin while he remains still, staying 1 meter away throughout. Then Quentin rotates 120° clockwise around Polly while she remains still, staying 1 meter away throughout. They repeatedly do this pair of moves from their new positions. How many meters from where she started is Polly after they have each moved 10 times? (A) 0 (B) √3/2 (C) 1 (D) √3 (E) 2

D. √3. Polly’s displacement vectors turn 240° between her moves, so each group of three sums to zero. The tenth is a unit-circle chord of length 2 sin(60°) = √3.

13. A jar contains 12 green marbles and 8 red marbles. In a game, Anika and Ben take turns drawing a random marble from the jar, with replacement. If Ben ever draws a red marble, he wins, and if Anika ever draws a green marble, she wins. Otherwise, the game continues with the next player’s turn. The probability that Ben eventually wins is A if Anika goes first and B if Ben goes first. What is B − A? (A) 2/25 (B) 4/25 (C) 4/19 (D) 6/19 (E) 9/20

D. 6/19. Conditioning on the first draw gives A = (2/5)B and B = 2/5 + (3/5)A. Thus A = 4/19 and B = 10/19.

14. A line segment is drawn between two points on the graph of y = x² whose x-coordinates have been chosen independently and uniformly at random in the interval [0, 2]. The probability that the y-coordinate of the midpoint of the segment is greater than 2 lies in which of the following intervals? (A) (0.20, 0.21) (B) (0.21, 0.22) (C) (0.22, 0.23) (D) (0.23, 0.24) (E) (0.24, 0.25)

B. (0.21, 0.22). The event x₁² + x₂² > 4 is the part of [0,2]² outside a radius-2 quarter circle. Its probability is 1 − π/4 ≈ 0.2146.

15. What is the value of log₂(sin(π/8)) + log₂(sin(3π/8)) + log₂(sin(5π/8)) + log₂(sin(7π/8))? (A) −3 (B) −2√2 (C) −2 (D) −3/2 (E) −√2

A. −3. Pair the supplementary angles after combining the logs. The product is [sin(π/8) cos(π/8)]² = 1/8.

16. Annie, Bobby, and Casey each receive an integer, not necessarily distinct, from 1 through 10. In how many ways can the numbers be given so that their sum is divisible by 7? (A) 139 (B) 140 (C) 142 (D) 143 (E) 145

E. 145. Count by how many values are from 8–10. With zero, one, or two, one of the 1–7 slots is determined modulo 7, giving 49, 63, and 27. With three, the residue patterns (1,3,3) and (2,2,3) give six more.

17. Belinda follows a path on the plane that is entirely contained within the square region 0 ≤ x, y ≤ 4. Call her path dignified if and only if all of the following conditions are satisfied. • Each step is one unit up, down, or to the right. • The path does not revisit any vertex. • The path starts at (a, 0) and ends at (b, 0), where a and b are integers with a < b, and touches the x-axis only at these two points. The following figure shows some examples of dignified and not dignified paths. How many dignified paths can Belinda follow? (A) 174 (B) 340 (C) 448 (D) 512 (E) 970

C. 448. For k right steps, there are 5 − k starting columns and 4^k choices of step heights. The heights force the vertical segments. Sum (5 − k)4^k for k = 1 through 4.

18. A ribbon wraps around a pole in such a way that equal amounts of area are covered and not covered by the ribbon. The width of the ribbon is 4 cm, and the circumference of the pole is 10 cm. The figure below shows a possible front view of a portion of the pole. In centimeters, how much of the pole height is needed for the ribbon to wrap around exactly once? (A) 15/2 (B) 8 (C) 10 (D) 25/2 (E) 40/3

E. 40/3. Unroll the pole. A turn of rise h has ribbon area 4√(100 + h²), half the area 10h of the pole per turn. Solve 4√(100 + h²) = 5h.

19. Starting with a list of three numbers, form a sequence in which each subsequent term is the average of the previous three terms. For example, starting with the sequence (2, 4, 9), the next two terms are (2 + 4 + 9)/3 = 5 and (4 + 9 + 5)/3 = 6. Which of the following initial three terms will produce a sequence that has the greatest 2026th term? (A) (1, 2, 3) (B) (2, 1, 1) (C) (3, 4, 8) (D) (4, 4, 7) (E) (4, 5, 6)

C. (3,4,8). C starts above A and B. C, D, and E tie at term 4, and C leads at terms 5 and 6. Averaging preserves the lead from term 7 on.

20. Which of the following statements most accurately describes the graph of the relationship sin(x²) = cos(y²)? (A) the set of all points (√(m/π), √((n + 1/2)/π)), where m and n are nonnegative integers (B) the set of all lines y = ±x + π/2 + kπ, where k is an integer (C) the set of all circles x² + y² = π/2 + 2πn, where n is a nonnegative integer (D) a set of infinitely many concentric circles and infinitely many hyperbolas (E) a set of finitely many ellipses centered at the origin and infinitely many parabolas

D. Infinitely many concentric circles and hyperbolas. Rewrite the cosine as sin(π/2 − y²). The two sine cases give x² + y² = π/2 + 2kπ or x² − y² = π/2 + 2kπ.

21. Let Tₙ be the nth triangular number, that is, T₁ = 1, T₂ = 1 + 2 = 3, T₃ = 1 + 2 + 3 = 6, and so on. The value of Σ(k = 1 to ∞) Tₖ · (1/3)ᵏ can be expressed as p/q, where p and q are relatively prime positive integers. What is p + q? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18

D. 17. The generating function for triangular numbers is x/(1 − x)³. At x = 1/3 it equals 9/8.

22. Rectangular card ABCD has AB = 1 and AC = 2. As shown in the figure, a second card, AEFG, congruent to the first, is stacked onto it so that A̅G̅ lies on A̅C̅ and the two cards share corner A. Additional cards are added, following the same pattern (with shared corner A), creating a fan of cards. The process stops when a new card fits exactly atop ABCD. The shape then created is a nonconvex polygon with rotational symmetry. What is the area of this polygon? (A) 20√3 − 24 (B) 34√3 − 48 (C) 6 + 3√3 (D) 12 (E) 8√3

A. 20√3 − 24. Each card rotates by 30°, giving 12 congruent sectors. With A = (0,0) and B = (1,0), one sector has outer corners (√3,1) and (1,√3), and their outer edges meet at (1,2 − 1/√3). The sector area is 5/√3 − 2.

23. For positive integers a and b, let R(a, b) denote the remainder when a is divided by b. Suppose N is the least positive integer such that R(N, 2) ≥ 1, R(N, 4) ≥ 2, R(N, 8) ≥ 4, …, R(N, 2²⁰⁰) ≥ 2¹⁹⁹. What are the last three digits of N? (A) 125 (B) 249 (C) 375 (D) 511 (E) 875

C. 375. Each condition forces one binary bit to be 1, so N = 2^200 − 1. Euler’s theorem gives N ≡ 0 (mod 125), and N ≡ 7 (mod 8). The unique residue modulo 1000 satisfying both is 375.

24. Let △ABC be a scalene triangle with incenter I. (Recall that the incenter is the intersection of the three angle bisectors.) The distances from I to the midpoints of the sides of △ABC are √11, √14, and √19. What is the sum of the possible areas of the circle inscribed in △ABC? (A) 10π (B) 44π/3 (C) 58π/3 (D) 88π/3 (E) 30π

A. 10π. For t = r², the offsets from the side midpoints to the tangency points are half the differences of vertex tangent lengths. The largest offset is the sum of the other two, so √(19 − t) = √(11 − t) + √(14 − t). The only feasible root is t = 10.

25. A finite integer tower is an expression of the form a₁^(a₂^(a₃^(…^aₙ))) where n ≥ 1 and a₁, a₂, …, aₙ are positive integers greater than 1. (Recall that exponentiation is performed from the top down. For example, 3^(2³) equals 3⁸, not 9³.) Two towers are considered different if they have a different number of layers (values of n) or at least one pair of numbers in corresponding positions differ. Define f(x) to be the number of different finite integer towers whose value is x. (For example, f(16) = 4 because 4 towers evaluate to 16, namely 16, 2⁴, 4², and 2^(2²).) What is f(2⁸¹⁹²)? (A) 17 (B) 32 (C) 42 (D) 43 (E) 47

D. 43. The count f(2^m) is 1 plus the sum of f(d) over divisors d > 1 of m, since the bottom base is 2^(m/d). As 8192 = 2^13, add 1 to the sum of f(2^j) for j = 1 through 13, which is 42.