In \(\Delta\)ABC below, AM, BN, and CP are concurrent at O. If BM = 1 , MC = 2 , CN = 3 , NA = 4 , and AP = 5 , what is the length of PB ?

 Let x and y be real numbers. What is the minimum value of the following function:

   \(f\left(x,y\right)=\sqrt{4+y^2}+\sqrt{\left(x-2\right)^2+\left(2-y\right)^2}+\sqrt{\left(4-x\right)^2+1}\)

Consider the set of all solutions (x,y) to the equation 3x² = 3y⁴ + 2013, where x and y are integers. What is the sum of all values of y occurring in such pairs (x,y) ?

The only way that 10 can be written as the sum of 4 different counting numbers is 1 + 2 + 3 + 4. In how many different ways can 15 be written as the sum of 4 different counting numbers?

Hannah gives clues about her six-digit secret number:
Clue 1: It is the same number if you read it from right to left.
Clue 2: The number is a multiple of 9.
Clue 3: Cross off the first and last digits. The only prime factor of the remaining four-digit number is 11.
What is Hannah’s six-digit number?

Kate took a math test that she missed due to sickness. Her perfect score of 100 points raises class average from 80 points to 81. How many students including her in the class took the test?

Let n be a four-digit integer whose digits are all distinct. If 9n equals the integer obtained by reversing the digits of n, compute n.

Find the units digit of \(999^{999^{999^{999}}}\)

How many integers between 1000 and 5000 are perfect squares?

Compute the sum of the digits of all multiples of 25 from 1 to 105 inclusive.

Compute the sum of all possible four-digit positive integers formed by using each prime digit exactly once.

A train to Washington, D.C. leaves from New York at noon and travels at a constant rate of 70 miles per hour. Also at noon, a train to New York leaves from Washington, D.C. along an adjacent and parallel track traveling at a constant rate of 40 miles per hour. They will meet each other somewhere along the route. How far away from each other will they be 1 hour before they meet?

The integer n, between 10000 and 99999, is abcde when written in decimal notation. The digit a is the remainder when n is divided by 2, the digit b is the remainder when n is divided by 3, the digit c is the remainder when n is divided by 4, the digit d is the remainder when n is divided by 5, and the digit e is the remainder when n is divided by 6. Find n. 

Let R and S be points on the sides BC and AC , respectively, of ΔABC , and let P be the intersection of AR and BS . Determine the area of ΔABC if the areas of ΔAPS , ΔAPB , and ΔBPR are 5, 6, and 7, respectively

Determine the smallest five-digit positive integer N such that 2N is also a five-digit integer and all ten digits from 0 to 9 are found in N and 2N

Two drivers began their journey with the same amount of petrol in their cars at the same time. The only difference is that the first driver's car could drive 4 hours in that amount of petrol and the second one could drive 5 hours. 
However, they only drove for some time and found that the amount of petrol that was left in one of the cars was four times the petrol left in the other one. 
For how long had they driven at this point of time?

20 mathematical questions were given to a student. For each correct question, the student was promised 10 chocolates. However, for every wrong question, it was decided that he would have to give back 5 chocolates.
To get maximum number of chocolates, the student answered all the 20 questions. As a result, he got some answers wrong and got 125 chocolates at the end of evaluation.
Can you calculate how many answers he gave were incorrect ?

Given a rectangle paper with a circle hole as the figure below. How to cut the paper with a line so that we have two parts with equal area.

Continuing the previous post:

Let N = 111...1222...2, where there are 1999 digits of 1 followed by 1999 digits of 2. Express N as the product of four integers, each of them greater than 1.