AMC 10 AMC 12 Preparation

2017 AMC 10/12 Workshop

Karen Ge

1 Algebra

A1. (2007B) How many non-congruent right triangles with non-zero integer leg lengths
have areas that are 3 times their perimeters?

(A) 6 (B) 7 (C) 8 (D) 10 (E) 12

A2. (2005B) All three vertices of an equilateral triangle are on the parabola y = x2,
and one of its sides has a slope of 2. The x-coordinates of the three vertices have
a sum of m/n, where m and n are relatively prime positive integers. What is the
value of m + n?

(A) 14 (B) 15 (C) 16 (D) 17 (E) 18

A3. (2008B) Let A0 = (0, 0). Distinct points A1, A√2, . . . lie on the x-axis, and dis-
tinct points B1, B2, . . . lie on the graph of y = x. For every positive integer n,
An−1BnAn is an equilateral triangle. What is the least n for which the length
A0An ≥ 100?

(A) 13 (B) 15 (C) 17 (D) 19 (E) 21

A4. (2004B) The graph of 2x2 + xy + 3y2 − 11x − 20y + 40 = 0 is an ellipse in the first

quadrant of the xy-plane. Let a and b be the maximum and minimum values of y
x

over all points (x, y) on the ellipse. What is the value of a + b?

(A) 3 √ (C) 7 (D) 9 √
(B) 10 2 2 (E) 2 14

A5. (2005A) Let P (x) = (x − 1)(x − 2)(x − 3). For how many polynomials Q(x) does
there exist a polynomial R(x) of degree 3 such that P (Q(x)) = P (x) ∗ R(x)?

(A) 19 (B) 22 (C) 24 (D) 27 (E) 32

A6. (2007A) Square ABCD has area 36, and AB is parallel to the x-axis. Vertices A, B,
and C are on the graphs of y = loga x, y = 2 loga x, and y = 3 loga x, respectively.
What is a?

√√√√
(A) 6 3 (B) 3 (C) 3 6 (D) 6 (E) 6

A7. (OMO Winter 2013) A permutation a1, a2, . . . , a13 of the numbers from 1 to 13 is
given such that ai > 5 for i = 1, 2, 3, 4, 5. Determine the maximum possible value of

aa1 + aa2 + aa3 + aa4 + aa5 .

(A) 42 (B) 45 (C) 51 (D) 53 (E) 55

1

(Karen Ge) 1 Algebra

A8. (NIMO 2013) The infinite geometric series of positive reals a1, a2, . . . satisfies

∞ −1 ∞ 1
2013 GM(a1, a2, . . . , an) = N
1= an = + + a1

n=1 n=1

where GM(x1, x2, . . . , xk) = √ · · · xk denotes the geometric mean. Compute
k x1x2

N.

(A) 20112 (B) 20122 (C) 20132 (D) 20142 (E) 20152

A9. (OMO Winter 2012) Let a, b, c be the roots of the cubic x3 + 3x2 + 5x + 7. Given
that P is a cubic polynomial such that P (a) = b + c, P (b) = c + a, P (c) = a + b,
and P (a + b + c) = −16, find P (0).

(A) 9 (B) 11 (C) 13 (D) 16 (E) 21

A10. (NIMO 2012) Select reals x1, x2, · · · , x333 ≥ −1, and let Sk = x1k + x2k + · · · + x3k33.
If S2 = 777, compute the least possible value of S3.

(A) 499 (B) 800 (C) 999 (D) 1110 (E) 1331

2

(Karen Ge) 2 Combinatorics

2 Combinatorics

C1. Find the number of distinct cubes that can be made by painting each face of a
given cube in one of the 5 given colors (not all of the colors have to be used). Cubes
are distinct if they cannot be obtained from each other using rotations.

(A) 720 (B) 760 (C) 800 (D) 960 (E) 1024

C2. (2014B) In a small pond there are eleven lily pads in a row labeled 0 through 10.

A frog is sitting on pad 1. When the frog is on pad N , 0 < N < 10, it will jump

to pad N − 1 with probability N and to pad N +1 with probability 1− N . Each
10 10

jump is independent of the previous jumps. If the frog reaches pad 0 it will be

eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond,

never to return. What is the probability that the frog will escape without being

eaten by the snake?

(A) 32 (B) 161 (C) 63 (D) 7 (E) 1
79 384 146 16 2

C3. (2003B) Three points are chosen randomly and independently on a circle. What is
the probability that all three pairwise distance between the points are less than the
radius of the circle?

1 1 1 1 1
(A) (B) (C) (D) (E)

36 24 18 12 9

C4. (2005B) Six ants simultaneously stand on the six vertices of a regular octahedron,
with each ant at a different vertex. Simultaneously and independently, each ant
moves from its vertex to one of the four adjacent vertices, each with equal probability.
What is the probability that no two ants arrive at the same vertex?

(A) 5 (B) 21 (C) 11 (D) 23 (E) 3
256 1024 512 1024 128

C5. (2009B) Parallelogram ABCD has area 1,000,000. Vertex A is at (0, 0) and all
other vertices are in the first quadrant. Vertices B and D are lattice points on
the lines y = x and y = kx for some integer k > 1, respectively. How many such
parallelograms are there?

(A) 49 (B) 720 (C) 784 (D) 2009 (E) 2048

C6. (2014A) The parabola P has focus (0, 0) and goes through the points (4, 3) and
(−4, −3). For how many points (x, y) ∈ P with integer coordinates is it true that
|4x + 3y| ≤ 1000?

(A) 38 (B) 40 (C) 42 (D) 44 (E) 46

3

(Karen Ge) 2 Combinatorics

C7. (2012A) Let S be the square one of whose diagonals has endpoints (0.1, 0.7) and
(−0.1, −0.7). A point v = (x, y) is chosen uniformly at random over all pairs of
real numbers x and y such that 0 ≤ x ≤ 2012 and 0 ≤ y ≤ 2012. Let T (v) be a
translated copy of S centered at v. What is the probability that the square region
determined by T (v) contains exactly two points with integer coordinates in its
interior?

(A) 0.125 (B) 0.14 (C) 0.16 (D) 0.25 (E) 0.32

C8. (Math Prize 2009) Consider a fair coin and a fair 6-sided die. The die begins with
the number 1 face up. A step starts with a toss of the coin: if the coin comes out
heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in
this step. After 5 such steps, what is the probability that the number 1 is face up
on the die?

(A) 35 (B) 35 (C) 1 (D) 37 (E) 37
216 192 6 216 192

C9. (OMO Winter 2013) Beyond the Point of No Return is a large lake containing

2013 islands arranged at the vertices of a regular 2013-gon. Adjacent islands are

joined with exactly two bridges. Christine starts on one of the islands with the

intention of burning all the bridges. Each minute, if the island she is on has at

least one bridge still joined to it, she randomly selects one such bridge, crosses it,

and immediately burns it. Otherwise, she stops. If the probability Christine burns

all the bridges before she stops can be written as m for relatively prime positive
n

integers m and n, find the remainder when m + n is divided by 1000.

(A) 113 (B) 114 (C) 115 (D) 116 (E) 117

C10. Find the number of rotationally distinct ways of coloring a tetrahedron’s edges
with three distinct colors.

(A) 80 (B) 87 (C) 96 (D) 105 (E) 115

4

(Karen Ge) 3 Geometry

3 Geometry

G1. (2011A) Triangle ABC has ∠BAC = 60◦, ∠CBA ≤ 90◦, BC = 1, and AC ≥
AB. Let H, I, and O be the orthocenter, incenter, and circumcenter of ABC,
respectively. Assume that the area of pentagon BCOIH is the maximum possible.
What is ∠CBA?

(A) 60◦ (B) 72◦ (C) 75◦ (D) 80◦ (E) 90◦

G2. (2006B) Isosceles ABC has a right angle at C. Point P is inside ABC, su√ch
that P A = 11, P B = 7, and P C = 6. Legs AC and BC have length s = a + b 2,
where a and b are positive integers. What is a + b?

(A) 85 (B) 91 (C) 108 (D) 121 (E) 127

G3. (2007B) Points A, B, C, D and E are located in 3-dimensional space with AB =
BC = CD = DE = EA = 2 and ∠ABC = ∠CDE = ∠DEA = 90o. The plane of

ABC is parallel to DE. What is the area of BDE?

√√ √√
(A) 2 (B) 3 (C) 2 (D) 5 (E) 6

G4. (2008B) Let ABCD be a trapezoid with AB||CD, AB = 11, BC = 5, CD = 19,
and DA = 7. Bisectors of ∠A and ∠D meet at P , and bisectors of ∠B and ∠C
meet at Q. What is the area of hexagon ABQCDP ?
√√√ √√
(A) 28 3 (B) 30 3 (C) 32 3 (D) 35 3 (E) 36 3

G5. (2012B) Square AXY Z is inscribed in equiangular hexagon ABCDEF √with X
on BC, Y on DE, and Z on EF . Suppose that AB = 40, and EF = 41( 3 − 1).

What is the side-length of the square?

√ √√ √
(A) 29 3 (B) 21 2 + 41 3 (C) 20 3 + 16
√ 2 2 √
√
(D) 20 2 + 13 3 (E) 21 6

G6. (2002B) A convex quadrilateral ABCD with area 2002 contains a point P in its

interior such that P A = 24, P B = 32, P C = 28, P D = 45. Find the perimeter of

AB C D.

√ √√
(A) 4 2002 (B) 2 8465 (C) 2(48 + 2002)

√ √
(D) 2 8633 (E) 4(36 + 113)

G7. (2015B) Four circles, no two of which are congruent, have centers at A, B, C, and

D, and points P and Q lie on all four circles. The radius of circle A is 5 times
8
5
the radius of circle B, and the radius of circle C is 8 times the radius of circle D.

Furthermore, AB = CD = 39 and P Q = 48. Let R be the midpoint of P Q. What

is AR + BR + CR + DR ?

(A) 180 (B) 184 (C) 188 (D) 192 (E) 196

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(Karen Ge) 3 Geometry

G8. (2015A) A collection of circles in the upper half-plane, all tangent to the x-axis,

is constructed in layers as follows. Layer L0 consists of two circles of radii 702
k−1
and 732 that are externally tangent. For k ≥ 1, the circles in j=0 Lj are ordered

according to their points of tangency with the x-axis. For every pair of consecutive

circles in this order, a new circle is constructed externally tangent to each of the

two circles in the pair. Layer Lk consists of the 2k−1 circles constructed in this way.
6
Let S = j=0 Lj , and for every circle C denote by r(C ) its radius. What is

1
?

C∈S r(C)

(A) 286 (B) 583 (C) 715 (D) 143 (E) 1573
35 70 73 14 146

G9. (OMO Fall 2012) In trapezoid ABCD, AB < CD, AB ⊥ BC, AB CD, and the

diagonals AC, BD are perpendicular at point P . There is a point Q on ray CA

past A such that QD ⊥ DC. Compute BP − AP given that
AP BP

QP AP 51 4
+=
AP QP 14 − 2.

(A) 3 (B) 47 √ √ (E) 57
14 (C) 14 (D) 2 7 14

G10. (NIMO 2012) In quadrilateral ABCD, AC = BD and B = 60◦. Denote by M
and N the midpoints of AB and CD, respectively. If M N = 12 and the area of
quadrilateral ABCD is 420, then compute AC.

(A) 36 (B) 37 (C) 38 (D) 39 (E) 40

G11. (NIMO 2013) Let AXY ZB be a convex pentagon inscribed in a semicircle of
diameter AB. Suppose that AZ − AX = 6, BX − BZ = 9, and that AY = 12 and
BY = 5. Find the greatest integer not exceeding the perimeter of quadrilateral
OXY Z, where O is the midpoint of AB.

(A) 21 (B) 22 (C) 23 (D) 24 (E) 25

6

(Karen Ge) 4 Number Theory

4 Number Theory

N1. (2010A) The number obtained from the last two nonzero digits of 90! is equal to n.
What is n?

(A) 12 (B) 32 (C) 48 (D) 52 (E) 68

N2. (2010B) For every integer n ≥ 2, let pow(n) be the largest power of the largest
prime that divides n. For example pow(144) = pow(24 · 32) = 32. What is the
largest integer m such that 2010m divides

5300

pow(n)?

n=2

(A) 74 (B) 75 (C) 76 (D) 77 (E) 78

N3. (2012A) Let f (x) = |2{x} − 1| where {x} denotes the fractional part of x. The
number n is the smallest positive integer such that the equation

nf (xf (x)) = x

has at least 2012 real solutions. What is n?

(A) 30 (B) 31 (C) 32 (D) 62 (E) 64

N4. (2013B) Bernardo chooses a three-digit positive integer N and writes both its base-5
and base-6 representations on a blackboard. Later LeRoy sees the two numbers
Bernardo has written. Treating the two numbers as base-10 integers, he adds them
to obtain an integer S. For example, if N = 749, Bernardo writes the numbers
10,444 and 3,245, and LeRoy obtains the sum S = 13, 689. For how many choices
of N are the two rightmost digits of S, in order, the same as those of 2N ?

(A) 5 (B) 10 (C) 15 (D) 20 (E) 25

N5. (2014B) The number 2017 is prime. Let S = 62 2014 . What is the remainder

k=0 k

when S is divided by 2017?

(A) 32 (B) 684 (C) 1024 (D) 1576 (E) 2016

N6. (2015A) For each positive integer n, let S(n) be the number of sequences of length
n consisting solely of the letters A and B, with no more than three As in a row and
no more than three Bs in a row. What is the remainder when S(2015) is divided
by 12?

(A) 0 (B) 4 (C) 6 (D) 8 (E) 10

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(Karen Ge) 4 Number Theory

N7. (2012B) Define the function f1 on the positive integers by setting f1(1) = 1 and if
n = p1e1p2e2 · · · pkek is the prime factorization of n > 1, then

f1(n) = (p1 + 1)e1−1(p2 + 1)e2−1 · · · (pk + 1)ek−1.

For every m ≥ 2, let fm(n) = f1(fm−1(n)). For how many N in the range
1 ≤ N ≤ 400 is the sequence (f1(N ), f2(N ), f3(N ), . . . ) unbounded? Note: A
sequence of positive numbers is unbounded if for every integer B, there is a member
of the sequence greater than B.

(A) 15 (B) 16 (C) 17 (D) 18 (E) 19

N8. (2014A) The number 5867 is between 22013 and 22014. How many pairs of integers
(m, n) are there such that 1 ≤ m ≤ 2012 and

5n < 2m < 2m+2 < 5n+1?

(A) 278 (B) 279 (C) 280 (D) 281 (E) 282

N9. (2004A) For each integer n ≥ 4, let an denote the base-n number 0.133n. The

product a4a5 · · · a99 can be expressed as m , where m and n are positive integers
n!

and n is as small as possible. What is the value of m?

(A) 98 (B) 101 (C) 132 (D) 798 (E) 962

N10. (OMO Fall 2012) Find the number of integers a with 1 ≤ a ≤ 2012 for which there
exist nonnegative integers x, y, z satisfying the equation

x2(x2 + 2z) − y2(y2 + 2z) = a.

(A) 1255 (B) 1256 (C) 1257 (D) 1258 (E) 1259
N11. (NIMO 2012) For how many positive integers 2 ≤ n ≤ 500 is n! divisible by 2n−2?

(A) 40 (B) 41 (C) 42 (D) 43 (E) 44
N12. (OMO 2013) For positive integers n, let s(n) denote the sum of the squares of the

positive integers less than or equal to n that are relatively prime to n. Find the
greatest integer less than or equal to

s(n)
n2 ,

n|2013

where the summation runs over all positive integers n dividing 2013.

(A) 668 (B) 669 (C) 670 (D) 671 (E) 672

8