Algebra
Combinatorics
Geometry
Number Theory
Random Competition Question
100
Given that (3x-2)^5=ax^5 + bx^4 + cx^3 + dx^2 + ex +f, compute a-b+c-d+e-f.
What is 3125?
100
Find the number of distinguishable permutations of the letter in the word MISSISSIPPI.
What is 34650?
100
Find the area of a triangle with side lengths 20, 21, and 29.
What is 210?
100
Factor 9951 completely.
What is 3*31*107?
100
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
What is 34? (AMC 10B 2010)
200
Express the sum 1^3 + 2^3 + ... + n^3 in closed form.
What is (n(n+1)/2)^2?
200
Find the number of unordered pairs (a,b) of 2-digit positive integers with their product a multiple of 4.
What is 2025?
200
In circle O, chords AB and CD intersect at R. If AR:RB = 1:4, and CR:RD = 4:9, find AB:CD.
What is 15/13?
200
Given 7,999,999,999 has at most two prime factors, find its largest prime factor.
What is 4,002,001?
200
What is the remainder when 2014^14 is divided by 100?
What is 16? (Mathleague Target Test Round 11414)
300
Determine the minimum value of 4m + 25/m.
What is 20?
300
A positive integer is said to bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. How many bi-digital numbers are there?
What is 243?
300
Instead of walking along two sides of a rectangular field, Fran took a shortcut along the diagonal, thus saving distance equal to half the length of the longer side. Find the length of the long side of the field, given that the length of the short side is 156 meters.
What is 208 meters?
300
For how many integers N between 1 and 2013 inclusive is the fraction (N^2 + 7)/(N+4) NOT in lowest terms?
What is 87?
300
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 n/11 is also a palindrome?
What is 11/30? (AMC 12A 2013)
400
If a^2 - 3a +1 = 0, compute (2a^5 - 5a^4 + 2a^3 - 8a^2)/1+a^2.
What is -13/3?
400
In the counting game Bing-Bong, Arlene starts counting at 1, but skips all multiples of 3 and all numbers that contain the digit 3. What is the 40th number in the sequence?
What is 76?
400
Circle O is tangent to lines AB and CD at A and C, respectively. The lines intersect and form a right angle at point P. There exists such a rectangle PETF such that T is tangent is to the circle and E and F lie on AP and PC, respectively. If PE = 9 and PF = 8, find the radius of the circle.
What is 29?
400
General Han's army has 1000 soldiers. However, on any given day, some of them might not show up. He orders them to line up in rows of 11, then in rows of 13, and finally in rows of 17, and each time he counts the number of soldiers not in a row. One morning, he finds that there are 3 soldiers left when the rest are in rows of 11, 4 soldiers left when the rest are in rows of 13, and 9 soldiers left when the rest are in rows of 17. How many soldiers are absent this morning?
What is 73?
400
In the accompanying figure, the outer square S has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
What is 750? (AIME II 2012)
500
Factor x^4 + 4y^4.
What is (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)?
500
How many factors does 20000000 have?
What is 72?
500
Quadrilateral CDEF is a parallelogram. Its area is 36 square units. Points G and H are the midpoints of sides CD and EF, respectively. Lines EG and DH meet line CF at I and J, respectively. What is the area of the triangle CDJ?
What is 36?
500
Find all possible values of integer m such that mx^2 - (m-2)x + m-3 = 0 has integer roots.
What is 1?
500
A large candle is 119 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 10 seconds to burn down the first centimeter from the top, 20 seconds to burn down the second centimeter, and 10k seconds to burn down the k-th centimeter. Suppose it takes T seconds for the candle to burn down completely. Then T/2 seconds after it is lit, the candle's height in centimeters will be h. Find 10h.
What is 350? (AIME II 2013)
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