Consider an array $A$ of length $n$. We can split $A$ into contiguous segments called pieces and store them as another array $B$ . For example , if $A = [1,2,3]$, we have the following arrays of pieces : $B = [(1),(2),(3)]$ contains three 1-element pieces . $B = [(1,2),(3)]$ contains two pieces,one having $2$ […]

Factor $ (a – b)^3 + (b – c)^3 + (c-a)^3$ by SYMMETRY. Okay, this is the problem. Let $f(a) = (a – b)^3 + (b-c)^3 + (c-a)^3$ obviously, if you let $a = b$ then, $f(b) = 0$, thus $(a – b)$ is a factor of $f(a)$. Then someone said : If $(a – […]

At the moment we are doing combinatorics and probability at school and it is a branch of mathematics that interests me probably more than anything else. Upon doing some of my own research I´ve come across a combinatorics problem from some past math contest and I am trying to find an efficient way to solve […]

As part of a preparatory course in the contest PUTNAM, I have to show $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$. I know that I can use the identity $\sum_{k=0}^n {n \choose k} {n \choose n-k}$ with the generating function $\sum_{k=0}^\infty \binom nk x^k = (1+x)^n$. However, I am not very aged (16 […]

In the book ‘Putnam and beyond’, page $173$, it has the following problem: Find the smallest cosntant $k>0$ such that $$\frac{ab}{a+b+2c} + \frac{bc}{b+c+2a} + \frac{ca}{c+a+2b} \leq k(a+b+c)$$ for every $a,b,c>0$. In the solution, it states that: Note that the inequality remains unchanged on replacing $a,b,c$ by $ta,tb,tc$ with $t>0$. Consequently, the smallest value of $k$ […]

Find all pairs of positive integers $(a,b)$ that satisfy $13^a+3=b^2$. If $a$ is even then $3=(b-13^{a/2})(b+13^{a/2})$ which have no solutions. Now if the case $a=2k+1$ is odd then $b^2-13.(13^k)^{2}=3$ I cant proceed from here, please help. Any other methods for the latter case?

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The […]

I’m creating a game and can’t seem to quite figure this out – driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn’t change. so when a new game starts it draws from the same test bank (the questions are not […]

Prove that there exist infinitely many Pythagorean integers $a²+b²=c²$ My key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but I fail at it. Other try is that $\sqrt{a^2+b^2}=c$ and so it is an equation of a circle, so I tried […]

I realize this is simply an A1 problem, but my proof seems way too simple, so I would like someone to point out whether or not it’s correct (and most importantly, fix any flaws in it). Problem. Suppose that a sequence $a_1, a_2, a_3, …$ satisfies $0 < a_n \leq a_{2n} + a_{2n+1}$ for all […]

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