A group of numbers \(A_1, A_2, \dots , A_{100}\) is created by somehow re-arranging the numbers \(1, 2, \dots , 100\).
100 numbers are created as follows: \[B_1=A_1,\ B_2=A_1+A_2,\ B_3=A_1+A_2+A_3,\ \dots ,\ B_{100} = A_1+A_2+A_3\dots +A_{100}.\]
Prove that there will always be at least 11 different remainders when dividing the numbers \(B_1, B_2, \dots , B_{100}\) by 100.
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
What time is it going to be in \(2025\) hours from now?
Prove that the product of five consecutive integers is divisible by \(30\).
Prove that if \(n\) is a composite number, then \(n\) is divisible by some natural number \(x\) such that \(1 < x\leq \sqrt{n}\).
The natural numbers \(a,b,c,d\) are such that \(ab=cd\). Prove that the number \(a^{2025} + b^{2025} + c^{2025} + d^{2025}\) is composite.
Prove that for an arbitrary odd \(n = 2m - 1\) the sum \(S = 1^n + 2^n + ... + n^n\) is divisible by \(1 + 2 + ... + n = nm\).
Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by \(24\).
Mattia is thinking of a big positive integer. He tells you what this number to the power of \(4\) is. Unfortunately it’s so large that you tune out, and only hear that the final digit is \(4\). How do you know that he’s lying?
You might want to know what day of the week your birthday is this year. Mathematician John Conway invented an algorithm called the ‘Doomsday Rule’ to determine which day of the week a particular date falls on. It works by finding the ‘anchor day’ for the year that you’re working in. For \(2025\), the anchor day is Friday. Certain days in the calendar always fall on the anchor day. Some memorable ones are the following:
‘\(0\)’ of March - which is \(29\)th February in a leap year, and \(28\)th February otherwise.
\(4\)th April, \(6\)th June, \(8\)th August, \(10\)th October and \(12\)th December. These are easier to remember as \(4/4\), \(6/6\), \(8/8\), \(10/10\) and \(12/12\).
\(9\)th May, \(11\)th July, \(5\)th September and \(7\)th November. These are easier to see as \(9/5\), \(11/7\), \(5/9\) and \(7/11\). A mnemonic for them is “9-5 at the 7-11".
Then find the nearest one of these dates to the date that you’re looking for and find remainders.
For example, \(\pi\) day, (\(14\)th March, which is written \(3/14\) in American date notation. It’s also Albert Einstein’s birthday) is exactly \(14\) days after ‘\(0\)’th March, so is the same day of the week - Friday in \(2025\).
What day of the week will \(25\)th December be in \(2025\)?