AMC 12
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?
A) 52, B) 55, C) 62, D) 68, E) 72
E) 72
What is the value of (11!-10!)/9!?
100
For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of cheese on top of the toast that is tangent to each side of the triangle. What is the area of this piece of cheese?
49pi/9
For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marsh-mallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat pieces and 9 marshmallow pieces, and exclaims, “This is such an unlucky bowl!” How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?
11
The largest prime factor of 101 101 101 101 is a four-digit number N. Compute N.
9901
The sum of two positive numbers is 5 times their difference. What is the ratio of the larger number to the smaller?
3/2
For what value x does 10^x*100^(2x)=1000^5?
3
Suppose rectangle FOLK and square LORE are on the plane such that RL = 12 and RK = 11. Compute the product of all possible areas of triangle RKL.
414
Tyler has an infinite geometric series with sum 10. He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler’s series.
1/5
Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order.
For example, 06/18/23 is such a date since 6 <18 <23, while today, 11/11/23, is not.
186
Integers x and y satisfy x > y > 0 and x + y + xy = 80. What is x?
26
Ximena lists the whole numbers 1 through 30 once. Emilio copies Ximena's numbers, replacing each occurrence of the digit 2 by the digit 1. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
103
Let ABCD be a rectangle with AB = 20 and AD = 23. Let M be the midpoint of CD, and let X be the reflection of M across point A. Compute the area of triangle XBD.
575
The formula to convert Celsius to Fahrenheit is
F◦= 1.8·C◦+ 32.
In Celsius, it is 10◦warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?
18 degrees
Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
6/7
What is the value of a for 1/log2(a) + 1/log3(a) + 1/log4(a) = 1?
a=24
The mean, median, and mode of the 7 data values 60, 100, x, 40, 50, 200, 90 are all equal to x. What is the value of x?
90
Let ABCD and WXYZ be two squares that share the same center such that WX ∥AB and WX <AB. Lines CX and AB intersect at P, and lines CZ and AD intersect at Q. If points P, W, and Q are collinear, compute the ratio AB/WX.
Chelsea goes to La Verde’s at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among n bags, so that each bag contains at most 13 pounds of fruit. What is the minimum n for which this is possible?
75
Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row.
11/6
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
47/256
Find the area of the shaded region.
(diagram)
6.5
A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1, 24, and 3, and the segment of length 24 is a chord of the circle. Compute the area of the triangle.
192
Suppose a, b, and c are real numbers such that
a2−bc= 14,
b2−ca= 14
c2−ab=−3.
Compute |a+b+c|.
17/5
There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.
5/108