3. (12 points) There are four weights of different masses. Katya weighs the weights in pairs. As a result, she got 1700, 1870, 2110, 2330, and 2500 grams. How many grams does the sixth weighing variant weigh?
Answer: 2090.

Solution. Let $x, y, z, t$ be the weight of each weight. Then the pairwise weighings will be $x+y, x+z, x+t, y+z, y+t, z+t$ grams. There are three pairs: 1) $x+$ y and $z+t, 2) x+z$ and 3) $y+t, x+t$ and $y+z$ with the same weight $x+y+z+t$. Let's find pairs with the same sum: $1700+2500=$
$1870+2330$. Notice that there are no other pairs of these numbers with the same sum. Then the sixth weighing option: $4200-2110=2090$.

Grading criteria. Full solution - 12 points. If the participant noticed that three pairs out of six numbers - the total weights of two weights, when summed, give the weight of all weights, and therefore must be the same, give 7 points. The total weight of all weights is found - another 2 points. Noted that there are no other pairs of these numbers with the same sum, except for the found ones - another 2 points. Arithmetic errors in the correct solution process - minus 2 points.