My linear class is weird.
Consider two continuous functions \(f,g\colon [0,1]\to\mathbb{R}\). Prove that \[\left|\int_{0}^{1}{f(x)g(x)\textrm{ d}x}\right|\leq\sqrt{\int_{0}^{1}{f(x)^2\textrm{ d}x}}\sqrt{\int_{0}^{1}{g(x)^2\textrm{ d}x}}.\] This problem is pretty trivial. Simply apply Cauchy-Schwarz to the Riemann sum approximation of the LHS and take the limit as \(n\rightarrow\infty\). This was a starred (challenge) problem in my linear homework following the week where we discussed Cauchy-Schwarz. Strange. Anyway, I've encountered a lot of cute problems lately. Among these is the following. Problem (HMMT): 10 students take a test. Each problem is solved by precisely 7 people and nine of the students solve exactly 4 of the problems. How many problems does the 10th student solve? Solution: Let the number of problems on the test be \(P\). Consider a set of \(P\) bins, with each corresponding to a particular problem, and a set of 10 bins, with each corresponding to a particular student. For every student that solves a certain problem, we place a stone in that problem's bin and for every problem that a student solves, we place a stone in that student's bin. Note that if we place a stone in a problem's bin, we must also place a stone in a student's bin. That is, there exists a bijection between the stones in the problem bins and the student bins. The total number of stones in the problem bins is given to be \(7P\) and the total number of stones in the student bins is \(x+4\cdot9\), where \(x\) is the number of problems that the 10th person solves. Hence, \[x=7P-36.\] Since \(x\geq0\), we must have \(P\geq6\). Furthermore, since the 10th person cannot solve more problems than are on the test, we have \(x=7P-36<P\), which means \(P\leq6\). Therefore, \(P=6\) and \(x=\boxed{6}\).
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Karel is a robot who exists on a two-dimensional grid. It is intended to provide a relevant scenario for first-time programming students to practice writing code and accomplishing tasks in such a way that they can visualize the results of the execution of their code. I am currently using Karel in my ECE 15 class.
Karel can,
My professor posed the following challenge problem in my class. Suppose Karel is at an arbitrary location in a row of an odd number of tiles, where each tile in the grid is empty and Karel has an infinite number of items in its bag. Find an algorithm that will bring Karel to the center of the row. My solution is to first travel to one of the ends of the row (say, left). Place an item, travel two tiles, place an item, and continue placing an item every two tiles. Since the number of tiles in the row is odd, you will complete this process by placing an item on the opposite end. Then, pick up this item, travel back to the other end, and move forward one tile (the first empty tile). Place the item there. Travel back to right end and come back towards the right until you meet the next item square. Pick up this item and bring it to the next side in the same process of traveling back to the left end and then moving right until you encounter an empty square. It is important to note that before picking up the items, we must check to see that the tile just beyond it is not occupied by an item. If it is, we have fully transferred half of the items in the row to the other side of the row, and we can terminate the process. In that case, the last square on which we dropped an item is the center square! I may implement this in C later. |
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