AMC 12 Math Test 2013 with Time Limit 25 Questions

Square ABCD has side length 10. Point E is on \overline{\rm BC} , and the area of ΔABE is 40. What is BE ?

A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

A flower bouquet contains pink roses, red roses, pink carnations, and red carna- tions. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

What is the value of

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy t dollars, and Dorothy gave Sammy d dollars. What is t − d ?

In a recent basketball game, Shenille attempted only three-point shots and two- point shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score?

The sequence S1, S2, S3, . . . , S10 has the property that every term beginning with the third is the sum of the previous two. That is,

S_n = S_n−2 + S_n-1 for n ≥ 3

Suppose that S₉ = 110 and S₇ = 42. What is S₄ ?

Given that x and y are distinct nonzero real numbers such that x + \sqrt{2x} , what is xy ?

Let S be the set of positive integers n for which \frac1n has the repeating decimal representation 0.\overline{\rm ab} = 0.ababab . . ., with a and b different digits. What is the sum of the elements of S ?

riangle ABC is equilateral with AB = 1. Points E and G are on \overline{\rm AC} and points D and F are on \overline{\rm AB} such that both \overline{\rm DE} and \overline{\rm FG} are parallel to \overline{\rm BC} . Furthermore, triangle ADE and trapezoids DF GE and F BCG all have the same perimeter. What is DE + FG ?

The angles in a particular triangle are in arithmetic progression, and the side lengths are 4, 5, and x. The sum of the possible values of x equals a + √b + √c, where a, b, and c are positive integers. What is a + b + c ?

Let points A = (0, 0), B = (1, 2), C = (3, 3), and D = (4, 0). Quadrilateral ABCD is cut into equal area pieces by a line passing through A. This line intersects \overline{\rm CD} at point (\frac{p}{q}, \frac{r}{s}) , where these fractions are in lowest terms. What is p + q + r + s ?

The sequence

log₁₂ 162, log₁₂ x, log₁₂ y, log₁₂ z, log₁₂ 1250

is an arithmetic progression. What is x ?

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cottontail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

A, B, and C are three piles of rocks. The mean weight of the rocks in A is 40 pounds, the mean weight of the rocks in B is 50 pounds, the mean weight of the rocks in the combined piles A and B is 43 pounds, and the mean weight of the rocks in the combined piles A and C is 44 pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C ?

A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^th pirate to take a share takes \frac{k}{12} of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the 12^th pirate receive?

Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?

In ΔABC, AB = 86, and AC = 97. A circle with center A and radius AB intersects \overline{\rm BC} at points B and X. Moreover \overline{\rm BX} and \overline{\rm CX} have integer lengths. What is BC ?

Consider

A = log (2013 + log (2012 + log (2011 + log(· · · + log (3 + log 2) · · ·)))).

Which of the following intervals contains A ?

A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome n is chosen uniformly at random. What is the probability that \frac{n}{11} is also a palindrome?

ABCD is a square of side length √3 + 1. Point P is on \overline{\rm AC} such that AP = √2. The square region bounded by ABCD is rotated 90^◦ counterclockwise with center P , sweeping out a region whose area is \frac{1}{c} (aπ + b), where a, b, and c are positive integers and gcd(a, b, c) = 1. What is a + b + c ?

Three distinct segments are chosen at random among the segments whose end- points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area