A ball \(A\) is placed on the pool table as shown below. The walls are labeled as follows: wall \(1\) is the top wall, wall \(2\) is the right wall, wall \(3\) is the bottom wall, and wall \(4\) is the left wall.
If we hit the ball so that it bounces off the walls labeled \(1\), \(2\), \(3\), and \(4\) —in that order—and then stops rolling just before hitting the next wall, what region of the pool table can the ball reach? If the ball hits a corner, it also stops.
A point \(A\) is placed on the bottom edge of a cylinder, along the rim of the lower circle. Another point \(B\) is placed on the top circle, but opposite to opposite to \(A\), so that \(A\) and \(B\) are as far away as possible from each other.
Find the shortest path along the curved surface of the cylinder that goes from \(A\) to \(B\).
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.
Hello! I have a trick for you. Think of a number, which we call the original number from now on. Do the following:
Add fifteen to the original number.
Multiply the resulting number by four.
Add eight times the original number to the new result.
Divide by six.
Subtract twice the original number.
But I already know the finally answer! How?
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?