Algebra
The smallest of three consecutive odd integers that sum to 45.
What is 13?
The area of a square with perimeter 48
What is 144 units^2
The number of distinct prime factors that 2((5+3)^2) has
The percentage of students in a 200 student school that prefer jazz music if 50 prefer rock, 70 prefer rap, 20 prefer classical, and the remaining prefer jazz
What is 30%?
The fractional probability of pulling a black ball twice in a row in a bag where there are 5 black balls, 3 green balls, 1 purple ball, and 12 gold balls
1/21
For each non-zero real number a, we define a* = 5/a. The expression (100*)* is equal to?
What is 100?
Point D lies on side BC of △ABC so that AB=AD=CD. If ∠ABC=80 degrees, what is the measure of ∠ACD?
What is 40 degrees?
Amr, Bai, Cindy, and Derek divide N coins such that:
Amr receives 1/3 of the total coins the others receive.
Bai receives 1/5 of the total coins the others receive.
Cindy receives 1/7 of the total coins the others receive.
If N<100, what is the largest possible value of N?
What is 48?
Yesterday, 200 people bought ice cream at the Coconut Creamery.
A total of 85 people ordered fudge with their ice cream.
60 people ordered sprinkles.
32 people ordered both.
How many of the 200 people ordered neither fudge nor sprinkles?
What is 87?
Farhan has a blue hat, a white hat, a blue scarf, a white scarf, and a green scarf. He randomly chooses one hat and one scarf. What is the probability that the hat and the scarf are the same colour?
What is 1/3?
The lines with equations y = mx + 7, y = 2, x = 0, and y = 0 form a trapezoid with area 3. If m > 0, what is the value of m?
What is 4?
The area of a right-angled triangle is 54 cm². The side lengths are a cm, b cm, and c cm (hypotenuse), where a<b<c are positive integers. What is the value of cc?
What is 15cm?
What is the hundreds digit of the smallest five-digit positive integer that is divisible by 12, 13, 14 and 15?
In a 2000 m race, Arturo, Morgan, and Henri run at constant but different speeds. Arturo finishes 200 m ahead of Morgan and 290 m ahead of Henri. If Morgan and Henri continue at their same speeds, how far ahead of Henri will Morgan finish?
What is 100m?
A tennis tournament starts with 8 players. Francesca is equally likely to play against any of the other 7 players in her first match. If Francesca plays against Dominique or Estella, the probability that Francesca wins is 2/5 . If Francesca plays against any of the other 5 players, the probability that she wins is 3/4 . What is the probability that Francesca wins her first match?
What is 13/20?
Determine all real numbers x for which
(log x)^ [log (logx)] = 10000
x=2^(−2/3) or x=8.
Geometry 400 question
80
What is the largest palindrome less than 200 that is the sum of three consecutive integers?
Suppose that a palindrome p is the sum of the three consecutive integers a−1, a, a+1.
In this case, p=(a−1)+a+(a+1)=3a, so p is a multiple of 3.
The largest palindromes less than 200 are 191, 181, 171.
Note that 191 and 181 are not divisible by 3, but 171 is divisible by 3. We can easily check this using the divisibility by 3 test. For each of these integers, the sum of their digits is 11, 10 and 9, respectively. Only 9 is divisible by 3 and so 171 is the only one that is divisible by 3.
171 （56+57+58）
In the multiplication shown, each of P, Q, R, S, and T is a digit. The value of P+Q+R+S+T is?
P Q R S T 4
X 4
= 4 P Q R S T
14
How many 5-person committees can be selected from six teachers and eight students if there must be at least two students included?
1876
In △ABC, AB=8, and ∠CAB=60°. Sides BC and AC have integer lengths a and b, respectively. Find all possible values of a and b.
(a,b)=(7,3),(7,5),(8,8),(13,15).
Given two circles, the line joining their points of intersection is called their common chord. It can be shown that the common chord is perpendicular to the line connecting the centres of the circles. (Can you prove this?) Given the circles x^2+y^2=4 and x^2+y^2−6x+2=0, find the length of their common chord.
2 sqrt 3
Charles was born in a year between 1300 and 1400. Louis was born in a year between 1400 and 1500. Each was born on April 6 in a year that is a perfect square. Each lived for 110 years. In what year while they were both alive were their ages both perfect squares on April 7?
Since 1300≈36.06^2, the first perfect square larger than 1300 is 37^2=1369. The next perfect squares are 38^2=1444 and 39^2=1521.
Since Charles was born between 1300 and 1400 in a year that was a perfect square, Charles must have been born in 1369.
Since Louis was born between 1400 and 1500 in a year that was a perfect square, Louis must have been born in 1444.
Suppose that on April 7 in some year, Charles was m^2 years old and Louis was n^2 years old for some positive integers m and n. Thus, Charles was m^2 years old in the year 1369+m^2 and Louis was n^2 years old in the year 1444+n^2.
Since these expressions represent the same years, we have that 1369+m^2=1444+n^2, or m^2−n^2=1444−1369=75. In other words, we want to find two perfect squares less than 110 (since their ages are less than 110) whose difference is 75.
The perfect squares less than 110 are 1,4,9,16,25,36,49,64,81,100. The two that differ by 75 are 100 and 25. Thus, m^2=100 and n^2=25.
This means that the year in which the age of each of Charles and Louis was a perfect square was the year 1369+100=1469.
A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, 3, 2, 4, 1, 6, 5 is a permutation of 1, 2, 3, 4, 5, 6. We can write this permutation as a1, a2, a3, a4, a5, a6, where a1=3, a2=2, a3=4, a4=1, a5=6, and a6=5. Determine the average value of a1−a2+a3−a4+a5−a6+a7 over all permutations a1, a2, a3, a4, a5, a6, a7 of 1, 2, 3, 4, 5, 6, 7.
4