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 2d comment Find the correct betting combination @AlexeyBurdin is it really 3,5,7,17,22 ? I mean, if you bet t/3 on the first, you will only have 2t/3 left if it wins. With a total bet of t, it's a loss. Or you mean you also recover your bet, but I thought it was the meaning of the first "Rs.1" in what you get. 2d comment Find the correct betting combination I guess $Rs$ means rupees (you may remove the money symbol, as it's useless, and is unclear in many countries). Is the gain proportional to the bet? I mean, if I bet $n$ rupees on team $5$ and I win, will I win $21n$ rupees? 2d comment What is the probability that 5 randomly chosen cards in a deck add up to 40 or greater? Thanks! I'm glad we finally get the same answer. :) 2d comment What is the probability that 5 randomly chosen cards in a deck add up to 40 or greater? I don't find the same answer, but I have a doubt :) Do you take the number of possibilities of combinations into account? I mean for the (1,1,1,2) case for instance, there are $5 \choose 2$ ways to place the pair. You don't seem to take the order into account at all (hence the 2598960), but I'm not sure it's correct, since the different "patterns" ((1,1,1,1,1), (1,1,1,2)...) don't have the same number of possibilities. 2d comment If $\det A=1$ and the matrices $A^{2015}$ and $A^{2017}$ are integer, is $A$ an integer matrix? Even simpler: $A^2$ is an integer matrix, so $A^{2016}=(A^2)^{1013}$ is also an integer matrix. No need for recursion, though it also works of course. Now check determinant, and apply the same trick as above to get $A^2$, and you'll find $A$. May25 comment Hessian-Matrix positive definite $\iff$ $a$ local minimum? Try $f(x,y)=x^4+y^4$ at $x=y=0$. May25 comment Fibonacci-related infinite sum For the first, this and this give a reference to an article of Edmund Landau which is here pp. 298-300 (in french). May25 comment Fibonacci-related infinite sum Not completely unrelated : mathoverflow.net/questions/51426/… May25 comment Inductively prove that any natural number $\ge 12$ can be written as the sum of 4s and 5s Notice that the set $S$ of numbers $n\geq12$ that cannot be written $n=4a+5b$ is either empty, either has a minimum element $n_0$ (since $S$ is then a nonempty set of nonnegative integers, or a subset of $\Bbb N$, which is well ordered). But then $n_0$ must be $\leq15$, otherwise one of $n_0-4$ or $n_0-5$ would be in $S$, contradiction. Thus you just have to check $12,13,14,15$. Almost the same argument will give you an induction proof. May25 comment Sum of zeros of $P(x)$ Statement of the problem (#10) and related discussion on Art of Problem Solving. Since you wrote the same question there, it's nice to tell :) May25 comment How does $\cos (2z) = e^{2zi}$? @snowman sine is odd May14 comment Probability of dice with a cumulative successes Do you mean that according to his skill, the attacker will roll 1, or 2 or .. or 5 dice? And for each outcome that is 9 or 10, he will roll another die? Then, does this apply to the additionnal dice (I mean, if he gets 9 or 10 on the 6th die, he will throw yet another die). Anyway, you have to know the "base" number of dice (between 1 to 5) to compute the probability to win. With only one die for example, the attacker can't win with more than 1 success, obviously. May14 comment Computing a certain $2014$-fold product using a particular associative binary operation $\ast$ @FlorinM. You also need the fact that it's commutative. Apr25 comment Convergence: infinite series Intuitive hint: the inequality means that if $b_n$ decreases, then $a_n$ decreases more, and if $b_n$ increases, $a_n$ increases less. So, apart form the first term, you should expect that the finite sums of $a_n$ are less than the finite sums of $b_n$. You can factor out the first terms without changing the successive quotients, to make this more precise. Apr25 comment Evaluation of $\int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos xdx$ Nice answer, and nice LaTeX lesson ;-) Apr25 comment How do I find the determinants of $3A, -A, A^2, A^{-1}$, where A is an $4\times 4$ matrix and $\det(A) = \frac{1}{3}$? For the (2), you can also use the fact that it's a multilinear form. Apr25 comment How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ This will be helpful: math.stackexchange.com/questions/544228/… Apr25 comment show that $(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$? Wrong. Try substituting $A =\pi/2$. Apr25 comment Finding pattern @Chou But then no answer leaves this property true, since the only primes are $2,3,11$, and they are all divisible by $5$. Apr25 comment For what types of differential equations is the Laplace transform most effective? Linear equations, especially (but not only) with constant coefficients. Laplace transform may be used for other types of equations, but it's less straightforward. See an example of the "easy" case here: math.oregonstate.edu/home/programs/undergrad/…