Algebra

Geometry

Counting

Number Theory

Random

100

If p^{3} - q^{3} is prime for prime numbers p and q, find the ordered pair (p, q).

(3, 2)

We can factor: p^{3} - q^{3} = (p - q)(p^{2} + pq + q^{2}). Since primes can only have two factors, (p - q) must be 1. The only possible primes for which this is true is 3

and 2. We have this because if p - q = 1, then one of p and q must be even.

100

A square has side-length 12. A new square is constructed by connecting the midpoints of the original square. Compute the area of the new square.

72

100

Pidge has 12 green socks, 14 yellow socks, and 9 orange socks. Pidge grabs the socks one at a time, at random. How many socks does Pidge need to grab to guarantee they have a pair of matching socks?

4

If Pidge picks one of each color sock, Pidge has 3 socks. If Pidge picks even one more sock, it will match one of the socks they already have, and they will have a matching pair.

100

What is the larger of two prime numbers with a sum of 2019?

2017

100

if n = (1,000,001 - 1)(1,000,001 +1), what is the least integer k such that n+k is a square number?

k = 1

200

Let (x; y) be a solution to the system

xy = 5

x^{2}y + xy^{2} + x + y = 54

Compute the value of x^{2} + y^{2}

71

200

In triangle TRI, point A is on RI such that RA = 5 and AI = 7. If the area of triangle TRI is 36, what is the area of triangle RAT?

15

200

How many ways are there to rearrange the letters in the word "fish"

24

consider 4 blank spaces to be filled with the setters of the word "fish"

_ _ _ _ ; f, i, s, h

There are 4 letters to choose from for the first space. Since one letter has now been chosen, there are 3 letters left to chose for the second. Then there are 2 choices for the third, and only one left for the fourth.

Then the number of ways to arrange the letters is

4*3*2*1 = 4! = 24

200

What is the smallest integer greater than 237 for which each digit of the integer is a prime number?

252

200

What is the least prime number that can be written as the sum of two composite numbers?

13

300

Seven distinct positive integers have mean 2020. Compute the greatest possible median of

these integers.

3532

Let the seven distinct integers be

0 < a < b < c < d < e < f < g.

Since their mean is 2020, we know that the sum of these seven are 2020 * 7. Additionally, their

median is d, so we want to maximize d, given that a + b + c + d + e + f + g = 2020 ?* 7. Take

a = 1; b = 2; c = 3 and maximize d using the fact that d < e < f < g. The numbers will be,

1, 2, 3, 3532, 3533, 3534, 3535

300

A cow is tied with a rope to the point (0, 0). Cows can't walk through walls. How much roaming space does the cow have if there is a square house with vertices at

(0, 5), (5, 5), (5, 0), (0, 0), and the rope is 6 feet long?

(55/2)*Pi

300

How many even 5 digit numbers are there?

(numbers cannot have leading zeros)

45000

The first digit can be anything but 0, the second, third, and fourth digit can be anything,

while the last digit has to be one of 0, 2, 4, 6, 8.

So, multiplying out the possibilities you

get 9*10*10*10*5 = 45000.

300

If N is a 3-digit integer, and the result of reversing the digits is the number M, what is the maximum value of N-M?

891

300

Compute the number of digits in the number

8^{673} *? 25^{1009}

2019

8 = 2^{3}, also 25 = 5^{2}

so, 8^{673} *? 25^{1009} = (2^{3})^{673} * (5^{2})^{1009}

= 2^{2019} * 5^{2018}

= 2 * 2^{2018} * 5^{2018}

= 2 * 10^{2018}

The number will be a 2 followed by 2018 zeros, for 2019 total digits.

400

A positive integer is called a dragon if it can be written as the sum of four (not necessarily distinct) positive integers a; b; c; and d such that

(a + 4) = (b - 4) = (4c) = (d/4)

Find the least dragon.

50

If a + 4 = 4c, we know that c > 1 since a is positive. If c = 2; then a = 4; b = 12; and d = 32 so the smallest dragon is 4 + 12 + 2 + 32 = 50:

400

ABCD is a rectangle with AB = 5 and AD = 12. Point P is on AD. E and F are feet of the altitudes from P to BD and AC, respectively. Compute PE + PF.

60/13

400

How many diagonals (line segments connecting two nonadjacent vertices) are in a do-decagon? (12-sided polygon)

54

A diagonal could connect a vertex to any other vertex except itself, or either of its two neighboring vertices. So, for each of the twelve vertices in a do-decagon, there are 12 - 3 = 9 vertices that a diagonal can connect it to.

There are 12*9 ways to draw a diagonal across the polygon. But this counts the same lines being drawn in both directions, so the actual answer is

(12*9)/2 = 54

400

How many different positive integers leave a remainder of 24 when divided into 9449?

9

400

Imagine a language where the longest word that exists is 5 letters long. If their alphabet only contains 4 letters, and every possible string of letters is a word, how many words exist in that language?

1364

Consider the different cases of words that are only 1 letter long, words that are two letters long, 3 letters, 4 letters, and 5 letters long.

For a word with 5 letters, there are 4 different possibilities for the first letter, 4 possibilities for the second, etc. Then the number of 5-letter words is

4*4*4*4*4 = 4^{5}

Similarly, the number of 4-letter words will be 4^{4}.

The numbers of 3-letter, 2-letter, and 1-letter words are, respectively, 4^{3}, 4^{2}, and 4.

Then the total number of words is

4^{5} + 4^{4} + 4^{3} + 4^{2} + 4 = 1361

500

A list consists of consecutive integers from -15 to x; inclusive. If the sum of all the integers in this list is 70; how many numbers are in this list?

35

We know that x > 0, since the sum of the list is positive, so we can find the sum of the numbers from -1 to -15, and if that sum is S = -n, we know that the sum of the numbers from 1 through x is

70 - S = 70 - (-n) = 70 + n. We can then find x, and our solution is x + 16.

500

From a point inside an equilateral triangle, the distance to the sides of the triangle are

2*sqrt(3), 4*sqrt(3), and 5*sqrt(3).

What is the area of the equilateral triangle?

500

How many ways are there to arrange the letters in the word "physics"

2520

Consider 7 blank spaces that are being filled with the letters of the word "physics"

_ _ _ _ _ _ _ ; p, h, y, s_{1}, i, c, s_{2}

There are 7 letter choices for the first spot. Since one letter was chosen, there are now 6 choices for the second spot. Then 5 for the third, 4 for the fourth, 3 for the fifth, 2 for the sixth, and only one choice left for the last.

So they can be arranged in 7*6*5*4*3*2*1 = 7! ways. However, some of these arrangements are the same.

"s_{1}s_{2}phiy" and "s_{2}s_{1}phiy" are both ssphiy. Each arrangement can be done in two different ways, so the actual number is (7!)/2 = 2520

500

What are all positive integers n, for which the least common multiple of n and 1000, is 2000?

16, 80, 400, 2000

500

How many permutations of 123456789 are multiples of 36? One such example is 234578916.

80,640

Since 36 = 4*9, to be divisible by 36, a number must be divisible by both 4 and 9.

To be divisible by 9, a numbers digits must add to a multiple of 9. All the digits from 1 to 9 add to 9 regardless of their order.

To be divisible by 4, there are last 2 digits must be divisible by 4. There are 22 2-digit numbers that are divisible by 4. 6 of them have a zero digit (20, 40, 60, 80). Two of them have repeating digits (44, 88). Since our number cannot have any zero digit, or repeating digits, there are 16 ways to arrange the last two digits of our number so that it is divisible by 4, and 36.

Finally, for each of these ways of arranging the last two digits, there are 7 remaining digits to arrange in the front. The number of permutations of 123456789 that are multiples of 36 is

16*7! = 80,640

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