Question 11. Martha multiplied two 2-digit numbers correctly on a piece of paper. Then she scribbled out three digits as shown. What is the sum of the three digits she scribbled out? We know that a 2-digit number ending with a 3 times 20-something will give us 300-something ending with a 2. Now, since we know that the first digit of the second number is a 2, we can figure out the first digit of the first number. If the digit was 2 or higher, then the multiplication would be at least 400, since 20 times 20 will give you 400. So the first digit must be 1. After this, the problem is simple. We know that the final solution has to end with a 2 and it has to be 13 times something. Now, the only way you can get a 2 at the end is if you multiply a 3 by a 4, since 3 times 4 is 12. So we get the solution 24. When we multiply 13 times 24, we get the answer of 312. If we add 1 plus 4 plus 1, we get 6. Therefore, the correct answer is B, 6. Question 12. A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew found the middle row of squares and colored it in. How many squares did he not color? Since we know that the rectangle has more than one row and that Andrew finds a middle row, we know that the rectangle will have to have an odd number of squares when looking at the rows. And since 40 is a special number, the only other odd number that it can be divided into is 5. So if we have an 8 by 5 area, and we can multiply 8 by 4, which is the amount of rectangles in the row, we get a total of 32. And the correct answer is C, 32. Question 13. Philip wants to know the weight of a book with half a gram precision. His weighing scales only weight in 10 gram increments. What is the smallest number of identical copies of this book that Philip should weigh together to be able to do this? The only way we can do this with half a gram of precision is to make the half a gram equal to 10 grams. So we must do 10 divided by a half, which gives us 20. So the answer is D, 20. Question 14. A lion is hidden in one of three rooms. A note on the door of room one reads, the lion is here. A note on the door of room two reads, the lion is not here. A note on the door of room three reads, 2 plus 3 equals 2 times 3. Only one of these sentences is true. In which room is the lion hidden? If we imagine three doors, we can immediately cross out D, D, D, we can immediately cross out door number three, since we know that 2 plus 3 is 5, and 2 times 3 is 6, and 5 does not equal 6. So door three is false. Now door one reads that the lion is here. If that is true, then door two says that the lion is not here. Then that must also be true by default. So door number one must be false, leaving door number two. Therefore, the correct answer is B, room two. Question 15. Valeriu draws a zigzag line inside a rectangle, creating angles of 10, 14, 33, and 26 degrees, as shown. What is the measure of angle theta? If we take a closer look at the rectangle with the zigzag, we can see more clearly. Now, knowing that a triangle has a total angle of 180, if we add all together, and knowing that this is a rectangle with all corners being 90 degrees, we can solve this problem. If we look at this first corner, then we can tell that it is 80 degrees. Since it is a 90 degree angle, and we know that 10 of it is in the other triangle, 90 minus 10 is 80. And since we have 14, we must do 180 minus 80 minus 14, which gives us 86. Now we can do this in the bottom corner as well. 90 minus 26 gives us 64. And once we know this, we have two angles of this triangle. So 64 and 33 must be subtracted from 180. We get 83. Now we know that a straight line has a total angle measure of 180. So to find theta, we must do 86 and 83 subtracted from 180. And that gives us theta. Therefore, theta equals 11. Correct answer is A, 11 degrees. Question 16. Alice wants to write down a list of prime numbers less than 100 using each of the digits 1, 2, 3, 4, and 5 exactly once and using no other digits. Which prime number must be in her list? We make a list of all the possible prime numbers with 1, 2, 3, 4, and 5. You see, you can make the list of 1, 2, 3, and 5. 4 is excluded because 4 on its own is not a prime number. We get 2, 5, and 13. 1, 2, and 53. And finally, 2, 3, 5, and 41. Now this is the only example where we're using all five digits only once. Therefore, there must be a 41 in the answer. So the correct answer is D, 41. Question 17. A hotel on an island in the Caribbean advertises by using the slogan 350 days of sun every year. According to this ad, what is the smallest number of days Willie Byrne has to stay at the hotel in 2018 to be certain of having two consecutive days of sun? Since we know that 2018 is not a leap year, we know that it will have 365 days. So we must take 365 and subtract it from 350, getting 15. We know that 15 days can be dark. So if we write out a list of days and we can see the black numbers are the ones with days without sun. If Willie Byrne has bad luck and every other day, starting with the second day, has no sun, then we know that he has to go through 1, 2, 3, 4, 5, and so on, seeing that every odd number day has sun and every even number day has darkness. And since there will be 15 days of darkness, once we get to the number 31, there will be no more dark days. So the next day will also be a bright day. Therefore, Willie Byrne has to stay at least 32 days to be sure he will have sun in two consecutive days. The correct answer is D, 32. Question 18. The diagram shows a rectangle and a line X parallel to its base. Two points A and B lie on X inside the rectangle. The sum of the area of the two shaded triangles is 10 centimeters. What is the area of the rectangle? On closer inspection of the rectangle, you notice that if we draw lines on points A and B going straight up and down like so, we can split the triangles in half and we notice that half of the rectangles that we've made is shaded and the other half is not in every example. Knowing this, we know that half the area of the rectangle is shaded while half of it is not. So to find the full area of the rectangle, we must multiply the shaded area by two, getting 20 centimeters. Therefore, the correct answer is B, 20 centimeters squared. Question 19. James wrote a different integer from 1 to 9 in each cell of the 3x3 table. He calculated the sum of the integers in each of the rows and in each of the columns of the table. Five of his answers are 12, 13, 15, 16, and 17 in any order. What is the sixth answer? To do this, we don't even have to draw out the table. We must only recognize the fact that if we add together all integers 1 through 9, we get a total of 45. And if we multiply this number, it is 90. If we do 90 minus all the answers, 90 minus 12 minus 13 minus 15 minus 16 minus 17, we get 17. So the correct answer is A, 17. Question 20. 11 points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is 2018. And the sum of all the distances between the second point and the other points, including the first one, is 2000. What is the distance between the first and second points? If we take the total of all the distances for the first part, 2018, and subtract it from 2000, we get the difference of the sums between the distances. So we do 2018 minus 2000, getting 18. And since we know that there are nine total sums between the first point and the other points, excluding the second point, the second point is closer than the first point to each of the other nine points by the same distance. The distance between the first point and the second point can be written using the equation 9x equals 18. If we simplify this by dividing both sides by 9, we get x equals 2. Therefore, the correct answer is B, 2.

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