[ Examples and counterexamples. Constructions ]

A first-grader has a hundred cards with natural numbers from 1 to 100 written on them, as well as a large supply of "+" and "=" signs. What is the maximum number of correct equations he can form? (Each card can be used no more than once, each equation can contain only one "=", and cards cannot be flipped or combined to form new numbers.)

#
In the equality, no less than three numbers are involved, so there cannot be more than 33 equalities. Let's show how to form 33 equalities.

25 equalities: $49+51=100, 47+52=99, \ldots, 1+75=76$. The remaining even numbers are from 2 to 50. Another 5 equalities: $18+30=48, 14+32=46, \ldots, 2+38=40$. The remaining numbers are: $4, 8, 12, 16, 20, 22, 24, 26, 28, 50$. Another 3 equalities: $4+12=16, 8+20=28, 24+26=50$.

## Answer

33 equalities.