7th and 8th Grade, September, 2019

    C.1249. Orhan bought a body height measuring tape, that you supposed to put up on the wall. One end of the tape has the “80 cm” mark on it, the other end has the “170 cm” mark on it. The person, whose height you want to measure, always stands on the floor, so supposedly you should be able to measure the heights of people between 80 cm and 170 cm tall. The tape is placed on the wall vertically, the bottom of it is where it is supposed to be, and all the division marks are evenly spaced. However, when we tested the tape for the first time and measured the height of the 130-cm tall Peter, the tape showed 120 cm at the top of his head, which means that the cm divisions are not where they are supposed to be. What is the height of the tallest person you can measure with this tape?

    C.1250. You make as many 4-digit numbers using each of the digits 1, 2, 3, and 4 exactly once. What fraction of all these numbers are divisible by 11?

    C.1251. ABCD is a square whose sides are 5 cm long. We draw the CDE equilateral triangle on the CD side with E outside the square.

    a) How big are the angles of triangle ABE?

    b) How long is the radius of the circumscribed circle of triangle ABE?

    C.1252. On the island of the triaces live three kinds of people: truthful people, who always tell the truth; liars, who always lie; and moody people, who tell the truth 2/3 of the time, otherwise they lie. We know that Sophia is moody, and that her brother, David, is not. To their grandma’s question David said: ” I do not like apricot jam.” Sophia replied immediately: “This is not even true!” What is the probability that David likes apricot jam?

    C.1253. I thought of a number. I added to it the sum of its digits, and I got 2019. What number did I think of?

    C.1254. I thought of a prime number. I multiplied it by 10 and added the original prime to it. I got a 4-digit number whose first two digits are 4 and zero. What prime number did I think of?

    C.1255. Take a rectangle and cut it by 6 straight lines parallel to its vertical sides and 6 straight lines parallel to its horizontal sides. The perimeter of each of the 49 small rectangles in centimeters is a positive whole number. Is it true that the perimeter of the original rectangle in centimeters is a whole number?

    C.1256. Using 5 identical cards, we created the following building. What is the sum of the two marked angles on the side view of this building?

    Please send your solutions here.

    The Sharma Kamala Educational Trust is sponsoring the participation of students from India. So, if you are a student living in India, please, send your solutions to: Group C from India

    Back to Paul Erdős International Math Challenge

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