Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 345. (October 2012)

K. 345. A total of 90 cm length of wire is used to make two pieces of the square frame with a handle that is illustrated in the figure. The length of the handle in cm is an integer, and it equals the length of the side of the square. The total area of the two frames is 170cm2. Find the difference between the heights of the two frames. (The height is measured from the bottom of the handle to the top of the square.)

(6 pont)

Deadline expired on November 12, 2012.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Legyen az egyik négyzet oldala \(\displaystyle x\), a másiké \(\displaystyle y\). Ekkor \(\displaystyle 5x+5y = 90\), és \(\displaystyle x^2 + y^2 = 170\). Olyan két pozitív egész számot keresünk, melyek összege 18, négyzetük összege 170. Az 1-17, 2-16, ... 9-9 számpárokat végignézve megoldásként csak a 7-11 páros adódik. Azaz az egyik keret 14 cm, a másik 22 cm magas, így a különbség 8 cm.


Statistics:

199 students sent a solution.
6 points:103 students.
5 points:44 students.
4 points:17 students.
3 points:18 students.
2 points:12 students.
1 point:3 students.
Unfair, not evaluated:2 solutions.

Problems in Mathematics of KöMaL, October 2012