Misleading or not?
A ball and a bat cost $1.10 in total. If the bat costs $1 more than the ball, how much does the ball cost?
$0.05
Prove that for every integer x, 2x is even.
2 is a factor of 2x, making it even no matter what x may be.
Define a prime number.
A prime number is a natural number greater than 2 where its only divisors are one and itself.
A polygon with six sides is called a:
hexagon
A runner runs 6 km in 30 mins. Find his average speed in km/h.
12 km/h
A baseball player throws a ball 100m straight up in the air. Assuming he catches the ball at the same height at which he'd thrown it, how much vertical distance did the ball travel?
200m
Let x and y be natural numbers. Prove that there exists a value x such that for all values of y, x-y is less than 3.
Suppose x is any number less than or equal to 3. Then, x-y will always be less than 3 because the minimum value of y is 1.
Find the largest number p such that p divides both 36 and 54.
18
A worker decides to climb a 12 vertical roof by leaning a ladder of length 13 m such that the top of the ladder is the same height as the top of the roof. How far away from the bottom of the building should he lean the ladder?
5 m
A bacteria colony doubles its size every day. A scientist notices that her petri dish with this colony was full on the 360th day. On which day was it half full?
Day 359
One train departs London for Glasgow at a speed of 100 km/h. Another train departs Glasgow for London at a speed of 200 km/h. When the trains meet, which train will be closer to London?
They will both be the same distance from London.
We can rewrite f(x) as (x2 - 4x + 4) + 5 = (x - 2)2 + 5 by completing the square. Since (x - 2)2 ≥ 0, then f(x) ≥ 5, and thus the minimum value of the function is 5.
Find the smallest integer p such that p(7!) is a perfect square.
Note: A factorial a! is the product of all natural numbers equal to or less than a. For example, 4! = 4 x 3 x 2 x 1.
We know that 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 24 x 32 x 5 x 7. A perfect square needs all of its prime factors raised to even exponents. For 7! to be a perfect square, it needs at least one more factor of 5 and 7. We take p = 5 x 7 = 35 to satisfy this condition.
The sum of the interior angles of a decagon is:
1440o
A driver aims to finish a 60 km trip in 60 mins. For the first half, he drives at a speed of 30 km/h. At which speed should he drive the second half so that he can still achieve his goal?
He cannot achieve his goal anymore because he already spent an hour on the first half of the trip.
Can you explain why this statement does not work?
"This statement is false".
This statement will contradict itself no matter what. If this statement is true, then it would be false. However, if the statement was false, then it will satisfy the statement "this statement is false", making it true.
Prove that 1,000,001 is not prime.
1,000,001 can be expressed as 1003 + 13. Factoring the sum of cubes, we get 1,000,001 = (100+1)(1002-(100)(1) + 12) = (101)(9901), proving that 1,000,001 is not prime.
Prove that if n is odd, then n2 is odd.
Let n be an arbitrary integer. Since n is odd, we can let n = 2k+1 for some integer k. Squaring this result gives us n2 = (2k+1)2 = 4k2+4k+1 = 2(2k2+2k) + 1. Since k is an integer, 2k2+2k is an integer as well. This then leaves us with the form 2(integer)+1, which results in an odd integer.
A regular hexagon is inscribed in a circle with radius 4. Find the area of the circle if the hexagon is removed.
16π - 16√3
Suppose that there are n lockers in a room and n students. Every nth student toggles every nth locker (toggling means opening a locker if it was originally closed, and vice versa). Determine a function L(n) that represents the number of lockers that remain open once all students have exited the room.
L(n) = ⌊√n⌋
A car is driving down a 5 km stretch of highway with a speed limit of 100 km/h. One speed camera clocks him at a speed of 95 km/h at the start of the highway stretch. Two minutes later, another camera clocks him at 100 km/h. Do the police have the right to ticket him for speeding, and how do you know?
Yes, the police do have the right to ticket him. The driver went through the 5 km stretch in only 2 minutes, suggesting that he was going at an average speed of 150 km/h. Since he entered the stretch of highway under a speed of 150 km/h, there must be a point where he went at least 150 km/h on that stretch of road.
Prove that there are no integer solutions (x,y) to the equation x2 - y2 - 10 = 0.
Add 10 to both sides to get the equation x2 + y2 = 10. Factoring gives (x+y)(x-y) = 10.
Now, if we pay attention to the factor pairs of 10, we either have (1,10), (2,5), or its negative counterparts. We can see that no matter which pair we pick, it will always consist of one odd and one even integer.
Now, if we look at (x+y) and (x-y), they must both be even or both be odd. This means that it is impossible to obtain a pair of (x+y) and (x-y) where one term is odd and the other term is even, and thus, there are no integer solutions to this equation.
Prove that there are infinitely many primes.
Next, let b = a+1. Now, every single prime that divides it will have a remainder of 1. Therefore, b is prime. Therefore, for any finite product of primes, there will always exist an integer i that is one greater than the product such that i is prime.
A rectangle has one side on the x-axis, one side on the y-axis, and its top-right vertex lies on the line y = -x + 2. Find the maximum possible area of this rectangle.
Let A be the area of the rectangle. We know that A = x(-x+2) = -x+2x = -x+2x-1+1 = -(x-1)2 + 1. Since -(x-1)2 ≤ 0, A ≤ 1. Therefore, the maximum possible area is 1.
Spot the following error in this following proof.
Let 2x = π+3. Then, (π-3)(2x) = (π-3)(π+3), which gives 2xπ - 6x = π2 - 9. Rearrange to get 9-6x = π2-2xπ.
Add x2 to both sides: 9-6x+x2 = π2-2xπ+x2. Then, (3-x)2 = (π-x)2, suggesting that 3-x = π-x and thus π=3.
The error comes after this step: (3-x)2 = (π-x)2. Note that √(x) returns |x|, not x. This means that we must consider the case where 3-x is negative. If this was the case, then x-3 = π-x, giving us 2x = π+3, which is what we originally defined.