Problem proposed by Bruno Holland and Samuel Vitosa
The numbers 1,2,3, …, 1000 are written on the board. Two players alternately erase one of the numbers on the board until exactly two numbers remain. If the sum of these numbers is divisible by 3, the first player wins, otherwise the second player wins. Can a player always guarantee victory no matter how the other plays?
Send your solution to [email protected] Don’t forget to write your name, city and school, if possible. We will publish the solution next Sunday.
Suppose x is the temperature at which the thermometer is at the same altitude. Note that at this temperature, the ratio between the x to zero and 100 to x distances on the second thermometer will be equal to the ratio between the x to 10 and 70 to x distances on the first thermometer. So,
(x-0) / (100-x) = (x-10) / (70-x)
Multiplied by the cross, we have:
70 x – x² = 100 x – x² – 1000 + 10 x
By canceling the x² term that appears on both sides of the equation, and rearranging the remaining terms, we have:
1000 = 40 x so x = 25
Therefore, at a temperature of 25 ° C, the heights of the marks on the thermometer will be the same.
Bruno Holland Professor of Federal University of Goiás and Samuel Vitosa Professor at the Federal University of Bahia.