a month. If a new boy or a new girl joined this class, one of these two conditions would no longer be true. How many children are there in the class? This problem tells us that there is one boy from each day of the week and one girl for each month in the year in the class. Since there are seven days in a week and 12 months in a year, we have seven boys plus 12 girls, which equals a total of 19 children. So our final answer is B. Problem number 12. In the diagram, the center of the top square is directly above the common edge of the lower two squares. Each square has sides with a length of 1. What is the area of the shaded region? One way to go about solving this problem is actually taking the figure and rotating it upside down, as I've shown here, and noticing that this shaded region is actually quite similar to a triangle with a rectangular portion being cut out. So if we can just find out the dimensions of a few of these side lengths, we might be able to calculate this very easily using the area formula for rectangles and triangles. So when we fill this in, we actually see that because the unit square here has a length of 1, that this parallel side will also be a length of 1. Also, it is given in the problem that the center of the square is rested directly above this common edge, meaning that 1 half of the side is right here, and 1 half of the side is over here. This means that we also have a unit of 1 half of the square exposed here, not being covered by the bottom square at all. So that means we have a segment of 1 half, and of course, the side length of the square, which is 1. Given all of these dimensions, we can first calculate the area of the original triangle, which would have a height of 2 and a base of 1 and 1 half, or 3 halves. And this is equal to 1 half times 3 halves times 2. That's the area of the triangle, this quantity right here. And then we would subtract the area of this rectangle, which is cut out, which is obviously just 1 times 1 half. And this is equal to 3 halves minus 1 half, which is equal to 1. And that's the area of the shaded region. So our final answer is C. Problem number 13. Every asterisk in the equation is to be replaced with either plus or minus so that the equation is correct. What is the smallest number of asterisks that must be replaced with plus? If we try to only replace one of the asterisks with a plus and the rest with minuses, then the largest possible number we can get for the pluses is 2 plus 5, which equals 7. Notice that because there's no asterisk in front of the first two, that this will always be a positive addition. And when we add up the minuses, well, we see that 2 plus 5 equals 7. And after adding up the minuses, we get a total of negative 17. And clearly, negative 17 and 7 are not equal. So when we put those together, we will not get 0. We want the value from the additions and the value from the subtractions to be equal. Notice that if we were to add 5 to 7 and take away a negative 5 from the 17, then we would get 12 and negative 12. And we can do this just by adding one more plus instead of a minus for one of the asterisks. In other words, 2 plus 5 plus 5 equals 12. And everything else subtracted equals negative 12. And 12 minus 12 certainly equals 0. So we only need two pluses in order to make the equation correct. Therefore, our answer is B. Problem number 14. During a rainstorm, 15 liters of water fell per square meter. How much did the water level rise in an outdoor pool? First, note some important conversions. One liter is equal to 1,000 cubic centimeters. And one square meter is equal to 10,000 square centimeters. Since 15 liters of water fell per square meter, this would mean that the water level rose by 15,000 cubic centimeters divided by 10,000 square centimeters. And this is equal to 1.5 centimeters. So our final answer is D. Question 15. A bush has 10 branches. Each branch has either five leaves only or two leaves and one flower. Which of the following could be the total number of leaves the bush has? If we let B be the number of branches with leaves only, then L be the total number of leaves. Then we get L is equal to 5B plus 2 10 minus B. So knowing this, we can simplify to 3B plus 20. Now, L minus 20 divided by 3 is equal to, simplified, L minus 2 3rds minus 6, since this is a counting number. But none of the answers, 45, 39, 37, and 31, are possible when you subtract by a fraction such as 20 divided by 3. So this task is impossible. The correct answer is E. None of A to D. Problem number 16. The mean score of the students who took a mathematics test was 6. Exactly 60% of the students passed the test. The mean score of the students who passed the test was 8. What was the mean score of the students who failed the test? To solve this problem, we're going to need to write out an equation. And this equation would be 0.6 times 8 plus 0.4 times question mark equals 6. The 0.6 signifies the 60% of the students that passed. And the 8 is their average score. And the 0.4 signifies the percentage of students that failed the test. The question mark is the score that they got on average. When we solve this equation, first, we have to multiply out the 0.6 by 8. And we get 4.8. And then we subtract that from both sides. This gives us 0.4 times question mark equals 1.2. Next, we want to divide both sides by 0.4, which would give us question mark equals 3. So our final answer is C. Problem number 17. One corner of a square is folded to the center of the square to form an irregular pentagon. The areas of the pentagon and of the square are consecutive integers. What is the area of the square? Notice that since the areas of the pentagon and square are consecutive integers, the difference in their area must be 1. This means that the triangle that was folded in to turn the square into a pentagon must have also had an area of 1. Furthermore, notice that we can split up the original square into eight such triangles by drawing lines. Since eight triangles make up the square, and the triangle has an area of 1, we know that 1 8th of the area of the square is 1, which means that the total area of the square must be 8. Therefore, our answer is C. Problem number 18. Rachel added the lengths of three sides of a rectangle and got 44 centimeters. Heather added the lengths of three sides of the same rectangle and got 40 centimeters. What is the perimeter of the rectangle? Let's call the sides of the rectangle x and y. In this case, we can write down a system of equations based on what's given in the problem. And this system of equations would be 2x plus y equals 44, and x plus 2y equals 40. If we want to find out the perimeter of the rectangle, what we can do, we can start by adding up the two equations, like so. And we get 3x plus 3y equals 84, or 3 times the quantity, x plus y equals 84. We know that the perimeter is equal to 2x plus 2y, or 2 times x plus y, which is just equal to 2 3rds of this value right here. So we know that 2 times x plus y is just equal to 2 3rds times 84. And this is equal to 56. So our final answer is B. Problem number 19. The diagram indicates the colors of some of the unit segments of a design. See the figure. Lewis wants to color each remaining unit segment in the design one of the following colors, red, blue, or green. Each of the six triangles must have one side of each color. What color can he use for the segment marked x? We're going to have to solve this diagram one side at a time. First, we mark down our given colors, the blues and the greens, originally presented in the problem. And now we start by noting that since we have this blue side right here for this triangle, we know that this side cannot be blue. And we know that since in this triangle we already have a green side, that this side can also not be green. Since it can neither be blue or green, we know that it must be red. Similarly, this side must also be red since it is adjacent to a blue and a green. So we mark these down. Next, we notice that we have two almost complete triangles. We have one that is green and red, which means that the last side here must be blue. And we have one that is red and blue, which means the last side must be green. So we mark those down. Next, we notice that this side right here is part of the triangle, which already has a blue, this triangle here. And this side is also part of a triangle, which already has a green, that is this triangle here. Which means that this side can neither be blue nor green, which means that it must be red. Since we now have an almost complete triangle with the x, we have one red side and one blue side. The last side can only be green. So our final answer is A. Problem number 20. Irina asked five of her students how many of the five had studied the day before. Paul said none. Bertha said only one. Alna said exactly two. Eugenie said exactly three. And Gerard said exactly four. Irina knew that those students who had not studied were not telling the truth. But those who had studied were telling the truth. How many of these students had studied the day before? Notice that because each of these students is giving a different answer, that means that only one of them can be telling the truth. And this is important because it tells us that only the person who gave the answer one is telling the truth. And that would be Bertha. So our correct answer is B. Notice that the answer cannot be zero because if the answer is zero, then Paul would be correct. However, that would mean that he studied. But since he said nobody studied, and at the same time it means that he did study, well, that's clearly a contradiction, which means it's not possible. So again, our correct answer is B, one.

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