There are some coins in a straight line, all showing heads. You can choose any coin to flip. When you flip one, your friend must flip the coins directly next to it (the ones on its left and right, if there are any). For example: if you flip the first coin, your friend only flips the second coin, and if you flip the second coin, your friend flips the first and third coins.
The question is: no matter how many coins there are, can we always make all of them show tails in the end?
The product of two positive numbers \(a\) and \(b\) is greater than \(100\). Prove that at least one of the numbers is greater than \(10\).
Write the contrapositive of the statement “If it is sunny outside, then I put on sunscreen and wear sunglasses”
What is the contrapositive of the statement: “If the temperature is above \(40^\circ\)C or below \(-10^\circ\)C, then it is not safe to go outside."
Some lines are drawn on a large sheet of paper so that all of them meet at one point. Show that if there are at least \(10\) lines, then there must be two lines whose angle between them is at most \(18^\circ\).
A whole number \(n\) has the property that when you multiply it by \(3\) and then add \(2\), the result is odd. Use proof by contrapositive to show that \(n\) itself must be odd.
Calculate the left side and the right side. \[2\times(12+3)\quad\quad 2\times 12 + 2\times 3\] \[3\times(0.8+1)\quad\quad 3\times 0.8 + 3\times 1\] \[(-2)\times (3-5)\quad\quad (-2)\times 3 + (-2)\times (-5)\] What do you notice?
Expand \((x_1+\dots + x_n)^2\) where \(x_1,\dots,x_n\) are real numbers.
Prove the Cauchy-Schwarz inequality \[(a_1b_1+\dots+a_nb_n)^2\leq (a_1^2+\dots+a_n^2)(b_1^2+\dots+b_n^2)\] where \(a_1,\dots,a_n,b_1,\dots,b_n\) are real numbers. If you already know a proof (or more!), find a new one.
Prove that there exist infinitely many natural numbers \(a\) with the following property: the number \(z = n^4+a\) is not prime for any natural number \(n\).