Mathematics
user587054
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8 Prove that if $a, b, c \in \mathbb{Z^+}$ and $a^2+b^2=c^2$ then ${1\over2}(c-a)(c-b)$ is a perfect square. Dec 7 '18
6 Prove that if $a,b,c \in \mathbb{R^+}\text{ and } abc=8\text{ then } {ab+4\over a+2}+{bc+4\over b+2}+{ca+4\over c+2}\ge6$ Dec 14 '18
6 Prove that $q^i \equiv 1 \pmod {n!}$ for all $q, n \in \mathbb{Z^+}$ where the prime factors of $q$ are greater than $n$ Oct 13 '18
6 If $a,b,c,d\in\mathbb{Z^+}$ where $ad=b^2+bc+c^2$, prove that $a^2+b^2+c^2+d^2$ is composite Mar 7 '19
5 Prove that for all $n\ge2\in\mathbb{Z}$ and $p$ is prime then $n^{p^p}+p^p$ is composite. Feb 8 '19
5 In $\triangle ABC$, if angle bisectors $AE$ and $CD$ meet at incenter $F$, and $|FE|=|FD|$, then the triangle is isosceles or $\angle B=60^\circ$ Aug 23 '19
5 If $a^2+b^2 \gt a+b$ and $a,b \gt 0$ Prove that $a^3+b^3 \gt a^2+b^2$ Sep 28 '18
4 Find all positive integral values of $x$ if $\prod_{m=0}^{1008} (x-{2m+1 \over 2})^{2m+1} \lt 0$ Sep 28 '18
4 Find $5$ numbers where the sum of all pairs gives the results $110, 112, 113, 114, 115, 116, 117, 118, 120, 121$ Dec 8 '18
4 Find all integer solutions for $2(x^2+y^2)+x+y=5xy$ Dec 31 '18
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