MathDB

2013 AIME Problems

Part of AIME Problems

Subcontests

(15)
11
2

Ms. Math's Playing Blocks

Ms. Math's kindergarten class has 1616 registered students. The classroom has a very large number, NN, of play blocks which satisfies the conditions:
(a) If 1616, 1515, or 1414 students are present, then in each case all the blocks can be distributed in equal numbers to each student, and (b) There are three integers 0<x<y<z<140 < x < y < z < 14 such that when xx, yy, or zz students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of NN satisfying the above conditions.

Number of $f$ for which $f(f(x))$ is constant

Let A={1,2,3,4,5,6,7}A = \left\{ 1,2,3,4,5,6,7 \right\} and let NN be the number of functions ff from set AA to set AA such that f(f(x))f(f(x)) is a constant function. Find the remainder when NN is divided by 10001000.
10
2

Sum of All Possible Sums of Roots of Cubic

There are nonzero integers aa, bb, rr, and ss such that the complex number r+sir+si is a zero of the polynomial P(x)=x3−ax2+bx−65P(x) = x^3 - ax^2 + bx - 65. For each possible combination of aa and bb, let pa,bp_{a,b} be the sum of the zeroes of P(x)P(x). Find the sum of the pa,bp_{a,b}'s for all possible combinations of aa and bb.

Maximize [BKL] where KL passes through a fixed point A

Given a circle of radius 13\sqrt{13}, let AA be a point at a distance 4+134 + \sqrt{13} from the center OO of the circle. Let BB be the point on the circle nearest to point AA. A line passing through the point AA intersects the circle at points KK and LL. The maximum possible area for △BKL\triangle BKL can be written in the form a−bcd\tfrac{a-b\sqrt{c}}{d}, where aa, bb, cc, and dd are positive integers, aa and dd are relatively prime, and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.
12
2

Triangle and Hexagon

Let △PQR\triangle PQR be a triangle with ∠P=75∘\angle P = 75^\circ and ∠Q=60∘\angle Q = 60^\circ. A regular hexagon ABCDEFABCDEF with side length 1 is drawn inside △PQR\triangle PQR so that side AB‾\overline{AB} lies on PQ‾\overline{PQ}, side CD‾\overline{CD} lies on QR‾\overline{QR}, and one of the remaining vertices lies on RP‾\overline{RP}. There are positive integers aa, bb, cc, and dd such that the area of △PQR\triangle PQR can be expressed in the form a+bcd\tfrac{a+b\sqrt c}d, where aa and dd are relatively prime and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.

Integer cubics with all roots of magnitude 20 and 13

Let SS be the set of all polynomials of the form z3+az2+bz+cz^3+az^2+bz+c, where aa, bb, and cc are integers. Find the number of polynomials in SS such that each of its roots zz satisfies either ∣z∣=20\left\lvert z \right\rvert = 20 or ∣z∣=13\left\lvert z \right\rvert = 13.
9
2

Folded Paper Equilateral Triangle

A paper equilateral triangle ABCABC has side length 1212. The paper triangle is folded so that vertex AA touches a point on side BC‾\overline{BC} a distance 99 from point BB. The length of the line segment along which the triangle is folded can be written as mpn\frac{m\sqrt{p}}{n}, where mm, nn, and pp are positive integers, mm and nn are relatively prime, and pp is not divisible by the square of any prime. Find m+n+pm+n+p. [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

Colorful tilings of a strip of length 7

A 7×17 \times 1 board is completely covered by m×1m \times 1 tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let NN be the number of tilings of the 7×17 \times 1 board in which all three colors are used at least once. For example, a 1×11 \times 1 red tile followed by a 2×12 \times 1 green tile, a 1×11 \times 1 green tile, a 2×12 \times 1 blue tile, and a 1×11 \times 1 green tile is a valid tiling. Note that if the 2×12 \times 1 blue tile is replaced by two 1×11 \times 1 blue tiles, this results in a different tiling. Find the remainder when NN is divided by 10001000.
13
2

Repeated Sequence of Triangles

Triangle AB0C0AB_0C_0 has side lengths AB0=12AB_0 = 12, B0C0=17B_0C_0 = 17, and C0A=25C_0A = 25. For each positive integer nn, points BnB_n and CnC_n are located on ABn−1‾\overline{AB_{n-1}} and ACn−1‾\overline{AC_{n-1}}, respectively, creating three similar triangles △ABnCn∼△Bn−1CnCn−1∼△ABn−1Cn−1\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}. The area of the union of all triangles Bn−1CnBnB_{n-1}C_nB_n for n≥1n\geq1 can be expressed as pq\tfrac pq, where pp and qq are relatively prime positive integers. Find qq.

Find the area of isosceles ABC

In △ABC\triangle ABC, AC=BCAC = BC, and point DD is on BC‾\overline{BC} so that CD=3⋅BDCD = 3 \cdot BD. Let EE be the midpoint of AD‾\overline{AD}. Given that CE=7CE = \sqrt{7} and BE=3BE = 3, the area of △ABC\triangle ABC can be expressed in the form mnm\sqrt{n}, where mm and nn are positive integers and nn is not divisible by the square of any prime. Find m+nm+n.
8
2
7
2

Area of Triangle formed by Centers of Faces

A rectangular box has width 1212 inches, length 1616 inches, and height mn\tfrac{m}{n} inches, where mm and nn are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of 3030 square inches. Find m+nm+n.

Clerks sorting files

A group of clerks is assigned the task of sorting 17751775 files. Each clerk sorts at a constant rate of 3030 files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar reassignment occurs at the end of the third hour. The group finishes the sorting in 33 hours and 1010 minutes. Find the number of files sorted during the first one and a half hours of sorting.
14
2

Sines, Cosines, and Powers of 2... Oh My!

For π≤θ<2π\pi\leq\theta<2\pi, let
P=12cos⁡θ−14sin⁡2θ−18cos⁡3θ+116sin⁡4θ+132cos⁡5θ−164sin⁡6θ−1128cos⁡7θ+… P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots and Q=1−12sin⁡θ−14cos⁡2θ+18sin⁡3θ+116cos⁡4θ−132sin⁡5θ−164cos⁡6θ+1128sin⁡7θ+… Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots so that PQ=227\tfrac PQ = \tfrac{2\sqrt2}7. Then sin⁡θ=−mn\sin\theta = -\tfrac mn where mm and nn are relatively prime positive integers. Find m+nm+n.

Sums of maximums of remainders

For positive integers nn and kk, let f(n,k)f(n,k) be the remainder when nn is divided by kk, and for n>1n>1 let F(n)=max⁡1≤k≤n2f(n,k)F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k). Find the remainder when ∑n=20100F(n)\displaystyle\sum_{n=20}^{100} F(n) is divided by 10001000.
6
2
15
2

# of Triples Satisfying Condition

Let NN be the number of ordered triples (A,B,C)(A,B,C) of integers satisfying the conditions
(a) 0≤A<B<C≤990\leq A<B<C\leq99, (b) there exist integers aa, bb, and cc, and prime pp where 0≤b<a<c<p0\leq b < a < c < p, (c) pp divides A−aA-a, B−bB-b, and C−cC-c, and (d) each ordered triple (A,B,C)(A,B,C) and each ordered triple (b,a,c)(b,a,c) form arithmetic sequences.
Find NN.

Cyclic trigonometric expression that dies to an identity

Let A,B,CA,B,C be angles of an acute triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}. \end{align*} There are positive integers pp, qq, rr, and ss for which cos⁡2C+cos⁡2A+2sin⁡Csin⁡Acos⁡B=p−qrs, \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, where p+qp+q and ss are relatively prime and rr is not divisible by the square of any prime. Find p+q+r+sp+q+r+s.
Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume ABCABC is obtuse with ∠B>90∘\angle B > 90^{\circ}.
5
2
4
2

Not Burnside's Lemma

In the array of 1313 squares shown below, 88 squares are colored red, and the remaining 55 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90∘90^{\circ} around the central square is 1n\tfrac{1}{n}, where nn is a positive integer. Find nn. [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100); [/asy]

Equilateral triangle at (1,0) and (2,2rt3)

In the Cartesian plane let A=(1,0)A = (1,0) and B=(2,23)B = \left( 2, 2\sqrt{3} \right). Equilateral triangle ABCABC is constructed so that CC lies in the first quadrant. Let P=(x,y)P=(x,y) be the center of △ABC\triangle ABC. Then x⋅yx \cdot y can be written as pqr\tfrac{p\sqrt{q}}{r}, where pp and rr are relatively prime positive integers and qq is an integer that is not divisible by the square of any prime. Find p+q+rp+q+r.
3
2

Two Squares and Two Rectangles

Let ABCDABCD be a square, and let EE and FF be points on AB‾\overline{AB} and BC‾\overline{BC}, respectively. The line through EE parallel to BC‾\overline{BC} and the line through FF parallel to AB‾\overline{AB} divide ABCDABCD into two squares and two non square rectangles. The sum of the areas of the two squares is 910\frac{9}{10} of the area of square ABCDABCD. Find AEEB+EBAE\frac{AE}{EB} + \frac{EB}{AE}.

Setting large candles on fire

A large candle is 119119 centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes 1010 seconds to burn down the first centimeter from the top, 2020 seconds to burn down the second centimeter, and 10k10k seconds to burn down the kk-th centimeter. Suppose it takes TT seconds for the candle to burn down completely. Then T2\tfrac{T}{2} seconds after it is lit, the candle's height in centimeters will be hh. Find 10h10h.
2
2
1
2

The AIME Triathlon

The AIME Triathlon consists of a half-mile swim, a 3030-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

Metric timekeeping

Suppose that the measurement of time during the day is converted to the metric system so that each day has 1010 metric hours, and each metric hour has 100100 metric minutes. Digital clocks would then be produced that would read 9:999{:}99 just before midnight, 0:000{:}00 at midnight, 1:251{:}25 at the former 3:003{:}00 \textsc{am}, and 7:507{:}50 at the former 6:006{:}00 \textsc{pm}. After the conversion, a person who wanted to wake up at the equivalent of the former 6:366{:}36 \textsc{am} would have to set his new digital alarm clock for A:BC\text{A:BC}, where A\text{A}, B\text{B}, and C\text{C} are digits. Find 100A+10B+C100\text{A} + 10\text{B} + \text{C}.