2016-2017 Math/Logic Contest
For grades 5 – 6
Solutions
1.
A train with a length of 20 meters passes a bridge in 4 seconds at a speed of 30 m/s. What is the length
of the bridge?
Answer: 100m
Solution.
S
E
Let L meters be the length of the bridge. Based on the diagram above and using the head of the train as
the reference point, we can see that the total distance the train travels to pass the bridge is (L + 20) m.
On the other hand, same distance can be calculated as speed multiplied by time:
,
which leads us to the answer L = 100m.
2.
Two numbers 3 and 9 are written on the board. You can write a new number on the board by adding
any two numbers that are already written on the board. Is it possible to use these operations to obtain
(a) number 30 on the board?
Answer: Yes
Solution. One way to obtain number 30 is as follows:
3, 9 - add 3 and 9, get 12;
3, 9, 12 – add 9 and 12, get 21;
3, 9, 12, 21 – add 9 and 21, get 30.
(b) number 50 on the board?
Answer: No
Solution. We can notice that both numbers initially written on the board are multiples of 3. By adding
two multiples of 3, we will always be getting another multiple of 3. That is why, at any moment all the
numbers on the board must be multiples of 3. However, number 50 is not a multiple of 3, and that is
why it cannot be obtained using given operations.
3.
Several consecutive (double-sided) pages fell out of the middle of an old book. The first page of the
section that fell out is page 251 and the last page number of the section that fell out is written with the
same digits as the first page but in a different order. How many pages are missing from the book?
Answer: 262 pages
Solution. We can make the following observations:
1) The number of the last page must be greater than 251. Since this number is written with digits 2, 5
and 1, we can conclude that the first digit of this number is 5 (it cannot start with 1, and 215 is less
than 251).
2) Since the pages are double-sided, the first and the last pages of the section must have different
parity. Hence, the only choice for the last digit of the last page number is 2.
It is now clear that the number of the last page is 512. The list of missing pages is then: 251, 252, …,
512, which can be rewritten as: 251, 251 + 1, 251 + 2, …, 251 + (512-251), or 251, 251 + 1, 251 + 2, …,
251 + 261. It is now easy to see that the total number of missing pages from the book is: 261 + 1 = 262.
4.
Is it possible to split 13 cakes equally among six students by cutting each cake into two or three equal
pieces (or not cutting it at all if unnecessary)?
Answer: Yes
Solution. We describe the cutting instructions below:
1) 6 cakes stay uncut;
2) 3 cakes are cut in half;
3) 4 cakes are cut into three pieces.
To summarize, we get 6 whole cakes, 6 halves and 12 thirds of a cake. Therefore, each student gets
equal amount of cakes: one whole cake, one half and two thirds.
5.
Five students, Nicole, Guy, Tomer, Nicholas, and Emma, competed in five contests, math, English, history,
geography, and physics. What place did each student achieve in each contest if:
 In each contest, there was a first, second, third, fourth, and fifth place.
 There were no ties in any of the contests.
 The student that got first place in math was last in all other contests.
 No student got more than two first places.
 Nicholas was third in the English contest. He was ahead of Tomer and Emma.
 Tomer was the best in geography but he was second in history.
 Nicole was the best in two contests in which Nicholas was third.
 Nicholas was the last in at least one contest.
 Tomer was fourth in one more contest than Nicholas.
 One student won one contest and was second in all other contests.
Nicole
Guy
Tomer
Math
English
History
Geography
Physics
Answer: see the competition results for each contest below
Nicholas
Emma
Math:
English:
History:
Geography:
Physics:
Emma, Guy, Nicole, Tomer, Nicholas
Nicole, Guy, Nicholas, Tomer, Emma
Guy, Tomer, Nicole, Nicholas, Emma
Tomer, Guy, Nicole, Nicholas, Emma
Nicole, Guy, Nicholas, Tomer, Emma