Inequality

by srnjbr, Mar 21, 2025, 4:32 PM

a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2

Graph Theory

by JetFire008, Mar 21, 2025, 4:22 PM

Prove that for any Hamiltonian cycle, if it contain edge $e$, then it must not contain edge $e'$.

Inspired by hunghd8

by sqing, Mar 21, 2025, 4:08 PM

Inspired by hunghd8

by sqing, Mar 21, 2025, 3:36 PM

Let $ a,b,c\geq 0 $ and $ a+b+c\geq 2+abc . $ Prove that
$$a^2+b^2+c^2-\frac{1}{2}a^2b^2c^2\geq 2$$$$a^2+b^2+c^2-abc-\frac{1}{2}a^2b^2c^2\geq \frac{3}{2}$$$$a^2+b^2+c^2- \frac{19}{10}abc-\frac{1}{2}a^2b^2c^2\geq -\frac{12}{25}$$$$a^2+b^2+c^2- \frac{3}{2}abc-\frac{1}{2}a^2b^2c^2\geq \frac{17\sqrt{17}-71}{16}$$
This post has been edited 1 time. Last edited by sqing, 2 hours ago

Prime for square numbers

by giangtruong13, Mar 21, 2025, 2:43 PM

Given that $a,b$ are natural numbers satisfy that: $\frac{a^3}{a+b}$ and $\frac{b^3}{a+b}$ are prime numbers. Prove that $$a^2+3ab+3a+b+1$$is a perfect squared number
This post has been edited 1 time. Last edited by giangtruong13, 3 hours ago

Find min

by hunghd8, Mar 21, 2025, 12:10 PM

Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$

Problem 4

by blug, Mar 15, 2025, 7:57 PM

In a rhombus $ABCD$, angle $\angle ABC=100^{\circ}$. Point $P$ lies on $CD$ such that $\angle PBC=20^{\circ}$. Line parallel to $AD$ passing trough $P$ intersects $AC$ at $Q$. Prove that $BP=AQ$.

Assisted perpendicular chasing

by sarjinius, Mar 9, 2025, 3:41 PM

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.

Interesting inequality

by sqing, Feb 26, 2025, 9:30 AM

a! + b! = 2^{c!}

by parmenides51, Mar 26, 2024, 3:44 PM

Determine all triples $(a, b, c)$ of positive integers such that
$$a! + b! = 2^{c!}.$$
(Walther Janous)

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