# 6. Option 1.

Nезнайка named four numbers, and Ponchik wrote down all their pairwise sums on six cards. Then he lost one card, and the numbers left on the remaining cards were $270, 360, 390, 500, 620$. What number did Ponchik write on the lost card?
Answer: 530.

Solution. Let the original numbers be $a \leq b \leq c \leq d$. Suppose the card with the maximum sum is lost. Then this sum is $c+d$. Therefore, $a+b=270$ and $a+b+c+d>270+620=890$. On the other hand, the sum of all numbers on the cards is $3a+3b+2c+2d=270+360+390+500+620=2140$. We get that $2140>1780+2c+2d$, hence $c+d<180$. This is a contradiction. Similarly, we can prove that the card with the smallest numbers was not lost. Therefore, the sum of all numbers on the cards is $270+620=890$. All pairwise sums can be divided into the following groups: 1) $a+b, c+d$, 2) $a+c, b+d$, 3) $a+d, b+c$. The sum of the numbers in each group is 890. This is only possible if the lost card has the number $890-360=530$, since 620 and 270, 390 and 500 form pairs.