Quant · Algebra

Inequalities and Modulus

3 sub-topics · 32 lessons · 85 questions · 15 PYQs

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Inequalities13 lessons · 33 questions · 5 PYQs

Modulus (Absolute Value)13 lessons · 39 questions · 9 PYQs

Floor and Fractional Part6 lessons · 13 questions · 1 PYQs

Formula sheet

Undoing the floor
$[x]=k\iff k\le x<k+1$
Multiplying an inequality by a negative number
$a>b,\ c<0\ \Rightarrow\ ac<bc$
Sum of non-negative terms equal to zero
$|u|+v^2+\sqrt{w}=0\iff u=v=w=0$
Quadratic positive for every real number
$ax^2+bx+c>0\ \forall x\iff a>0,\ b^2-4ac<0$
Lattice points on a diamond
$\#\{(x,y)\in\mathbb{Z}^2:|x-a|+|y-b|=k\}=4k$
Integer part and fractional part
$x=[x]+\{x\},\qquad [x]\in\mathbb{Z},\ 0\le\{x\}<1$
Bounds of the sum of pairwise products
$-\tfrac{1}{2}(a^2+b^2+c^2)\le ab+bc+ca\le a^2+b^2+c^2$
Sum of floor of square roots
$\sum_{k=1}^{(m+1)^2-1}[\sqrt{k}]=\sum_{j=1}^{m}j(2j+1)$