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Suppose each of x and y is a real number > 0

Find the minimum value of:

 
2x2+2y2+3xy+1
-------------
     x+y

Find the smallest whole number N such that N contains all of the digits from 0 through 9, and N² contains all of the digit pairs 00, 11, 22, ..., 99.

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by 4.

Arrange the integers from 1 to N in an order such that the sum of any two consecutive terms is a power of 2.

For what values of N do solutions exist?

Solve:
FOR = I x WAS
where W, A, and S represent consecutive digits.

Find all positive integers x such that ⌊x/5⌋-⌊x/7⌋=1.

The sides of a triangle are three consecutive integers and its inradius is 4. Find the circumradius.

What are the next two numbers in the following sequence ?
2,12, 2122, 1132, 211213, 312213, 212223, .........

A positive integer N contains each 2-digit combination exactly once:
00, 01, ..., 99.

(A) What is the smallest number of digits N could have?
(B) What is the largest number of digits N could have?
(C) What is the smallest possible value of N?
(D) What is the largest possible value of N?

(no leading zeros)

Consider a sequence generated by positive integer x where each term is of the form ⌊(2+√7)^x⌋. Prove that the sum of two consecutive terms will always be odd.

What is the largest possible area of a quadrilateral with sides 1,4,7,8?

Determine the quotient and the remainder when a positive integer N constituted entirely by 2024 sevens is divided by 10001.

When n=8, the expression ab(aⁿ-bⁿ) is divisible by 30 for all positive integers a and b, and 30 is the greatest such divisor.

Find a positive integer n such that the greatest common divisor of ab(aⁿ-bⁿ) for all positive integers a and b is n.

Solve the equation

(x²+3x-4)³+(2x²-5x+3)³=(3x²-2x-1)³.

Refer to Mileage.

What pair of palindromic readings could they have so that it will take the most miles until they are both palindromes again? Is there any reading they could have so that they will NEVER both become palindromes?

What is the largest square (not ending in 0) whose digits are in non-increasing order, for example 441 and 7744?

M is an n-digit positive integer composed of a string of random decimal digits (no leading zero) selected from a uniform distribution. For each case below, determine the value of the smallest number of digits associated with a probability of at least 0.5 that M contains:
(A) all of the digits 0 to 9
(B) all of the 2-digit combinations 00, 01, ..., 99
(C) all of the 3-digit combinations 000, 001, ..., 999

Given that:

    n
Sn= Σ k/(k+1)!
   k=1

Find the value of:

1-S2022
--------
1-S2023

Is there any integer multiple of N=3^2024 that includes no zeroes in its decimal representation?

Provide adequate reasoning for your assertion.

Two workers were given the job of making a batch of a certain car part. After the first worker had worked for 2h, and the second for 5h, they realised they had only completed half the task. Then after working together for another 3h, they calculated that they still had 5% of the whole task to complete. How long would it take each of them to complete the task, if they worked separately?

Find ordered pairs (x, y) that solve the equation

(x²-4x+7)(y²+2y+6)=15

N is a positive integer such that N! expressed in decimal contains:
(A) all of the digits 0 to 9
(B) all of the 2-digit combinations 00, 01, ..., 99
(C) all of the 3-digit combinations 000, 001, ..., 999

Find the smallest value of N in conformity with the given conditions.

If

VOICES+NOISES=SWEDEN

then what is

97R5148456 ?

No software, please….

The sum of the squares of four consecutive integers is two less than a square number.

What is the mean of the four integers?

Show that it is possible to fill a rectangular box with rectangular blocks so that:

(1) the dimensions of all of the blocks are integers,
(2) no edge of a block coincides with an entire edge of the box, and
(3) no two blocks have identical faces (that is, you could not have blocks of size A×B×C and A×B×D).

Find the box of least volume for which this is possible.

Rob drew four right-angled triangles. The hypotenuse of his first triangle was also the shortest side of his second triangle; the hypotenuse of his second triangle was also the shortest side of his third triangle; the hypotenuse of his third triangle was also the shortest side of his fourth triangle. The length in millimetres of each side of each triangle was an integer less than 100.

What were the lengths of the shortest and the longest sides that Rob drew?

Note: Adapted from Enigma number 1779 which appeared in New Scientist on 2013.

A pentagon has vertices A, B, C, D and E, where ABCE is a square of side length 2 units, and CDE is an equilateral triangle.

Points A, B and D lie on the circumference of a circle G.

What is the exact area of the part of circle G that lies outside pentagon ABCDE?

Find all possible real values of x that satisfy this equation:

xx = 4x+4