Algebra
What is the average of ⅙ and ⅛?
7/48 (2019 Fermat #9)
A larger cube has volume 64cm^3. A smaller cube has edges that are half the length of the edges of the larger cube. What is the volume of the smaller cube?
8cm^3 (2016 Gauss #13)
How many integers between 100 and 300 are multiples of both 5 and 7, but are not multiples of 10?
3 (2021 Pascal #10)
Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total?
30 (2016 Cayley #10)
I bought a new plant for my garden. Anika said it was a red rose, Bill said it was a purple daisy, and Cathy said it was a red dahlia. Each person was correct in stating either the colour or the type of plant. What was the plant that I bought?
Red daisy (2011 Gauss #10)
At the Lacsap Hospital, Emily is a doctor and Robert is a nurse. Not including Emily, there are five doctors and three nurses at the hospital. Not including Robert, there are d doctors and n nurses at the hospital. What is the product of d and n?
12 (2012 Pascal #10)
The coordinates of three of the vertices of a parallelogram are (0,0), (1,4), and (4,1). What is the area of this parallelogram?
15 (2012 Fermat #19)
How many positive whole numbers, including 1, divide exactly into both 40 and 72?
4 (2007 Gauss #15)
A gumball machine that randomly dispenses one gumball at a time contains 13 red, 5 blue, 1 white, and 9 green gumballs. What is the least number of gumballs that Wally must buy to guarantee that he receives 3 gumballs of the same colour?
8 (2011 Cayley #16)
While exploring a cave, Carl comes across a collection of 5-pound rocks worth $14 each, 4-pound rocks worth $11 each, and 1-pound rocks worth $2 each. There are at least 20 of each size. He can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
50 (2018 AMC 12A Problem 2)
Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton. Bruce drove at 50 km/h. Anca drove at 60 km/h, but stopped along the way to rest. They both arrived at Staton at the same time. For how long did Anca stop to rest?
40 (2015 Fermat #10)
A circle is centered at O, AB is a diameter and C is a point on the circle with angle COB = 50 degrees. What is the degree measure of angle CAB
25 (2010 AMC 10B #6)
The integer m is a perfect cube exactly when it is equal to n³ for some integer. For example, 1000 is a perfect cube since 1000 = 10³. What is the smallest positive integer k for which the integer 2⁴×3²×5⁵×k is a perfect cube?
60 (2021 Pascal #16)
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. How many different-looking arrangements are possible?
15 (2019 Pascal #19)
How many swaps does it take to sort the list of integers [100, 99, …, 3, 2, 1] in increasing order by only swapping adjacent numbers?
4950 (CCC '05 S5 - Pinball Ranking (simplified))
Chris received a mark of 50% on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered 25% of the remaining questions on the test correctly. If each question on the test was worth one mark, how many questions in total were on the test?
32 (2016 Pascal #19)
The vertices of an equilateral triangle lie on a circle with radius 2. Find the area of the triangle.
3√3 (2019 Fermat #20)
For how many pairs (m,n) with m and n integers satisfying 1 ≤ m ≤ 100 and 101 ≤ n ≤ 205 is 3m + 7n divisible by 10?
2625 (2020 Cayley #20)
Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins?
21/32 (2015 Fermat #21)
Find the minimum integer N such that N! is a multiple of 29 * 52.
12 (CCO Training - Least Multiple)
The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way?
478661 (2019 Pascal #22)
In the cube ABCDEFGH with opposite vertices C and E, J and I are the midpoints of segments FB and HD, respectively. Let R be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is R2?
3/2 (2018 AMC 8 #24)
If n is a positive integer such that 2n has 28 positive divisors and 3n has 30 positive divisors, then how many positive divisors does 6n have?
35 (1996 AHSME #29)
Nylah has her living room lights on a timer. Each evening, the timer switches the lights on randomly at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. For example, the lights could be switched on at exactly 7:30 p.m. and off at any one of the infinite number of possible times between 11 p.m. and 1 a.m. On a given night, Nylah's lights are on for t hours. What is the probability that 4 < t < 5?
7/20 (2015 Cayley #23)
A positive integer is called palindromic if its digits are the same forwards and backwards (eg. 1221, 22, 9). Find the 100th smallest palindromic integer.
919 (ABC363D - Palindromic Number)