Mathematical and Physical Journal
for High Schools
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Problem K. 327. (February 2012)

K. 327. The sum of four positive integers is 125. If the first number is increased by 4, the second number is decreased by 4, the third number is multiplied by 4 and the fourth number is divided by 4, the results will be all equal. What may be the four original numbers?

(6 pont)

Deadline expired on March 12, 2012.

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

Megoldás. A négy keresett szám legyen \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) és \(\displaystyle d\). Tudjuk, hogy \(\displaystyle a+b+c+d= 125\). Az elvégzett átalakítások során kapott egyenlőségek: \(\displaystyle a + 4 = b – 4 = c\cdot 4 = d/4\). Innen \(\displaystyle a\) segítségével kifejezhetjük a többit: \(\displaystyle b=a+8\), \(\displaystyle c=a/4+1\) és \(\displaystyle d=4a+16\), azaz összegük \(\displaystyle 125=a+a+8+a/4+1+4a+16\), ahonnan \(\displaystyle a = 16\). Tehát a négy eredeti szám: \(\displaystyle a = 16\), \(\displaystyle b = 24\), \(\displaystyle c = 5\) és \(\displaystyle d=80\).


209 students sent a solution.
6 points:185 students.
5 points:6 students.
4 points:3 students.
3 points:4 students.
2 points:4 students.
1 point:1 student.
0 point:4 students.
Unfair, not evaluated:2 solutions.

Problems in Mathematics of KöMaL, February 2012