}

For which $n>3$ can a set of weights with masses $1,2,3, \ldots, n$ grams be divided into three equal-mass piles?

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Six weights, whose masses are six consecutive numbers, can easily be divided into three equal mass piles.

## Solution

Let's call a number $n$ good if a set of weights with masses $1,2,3, \ldots, n$ can be divided into three equal mass piles. For a good $n$, the total mass of the weights, which is $1 / 2 n(n+1)$, is divisible by 3. Therefore, $n$ can only give remainders of 0 or 2 when divided by 3.

If the number $n$ is good, then the number $n+6$ is also good, since the weights with masses $n+1, n+2, \ldots, n+6$ can easily be divided into three equal mass piles.

It is easy to check that the numbers 5, 8, 9 are good: $1+4=2+3=5 ; 1+2+3+6=4+8=5+7 ; 1+2+3+4+5=6$ $+9=7+8$. That 6 is a good number is shown above. Therefore, good numbers are all numbers of the form $3 k(k \geq 3)$ and $3 k+2(k \geq 1)$.

## [Inscribed and Circumscribed Circles] Problem $\underline{35074}$ Topics: [ Midline of a Trapezoid ]  [ Systems of Linear Equations ]

Difficulty: $3+$

Grades: $9,10,11$

Inside triangle $A B C$, there are points $P$ and $Q$ such that point $P$ is at distances 6, 7, and 12 from the lines $A B, B C, C A$ respectively, and point $Q$ is at distances 10, 9, and 4 from the lines $A B, B C, C A$ respectively. Find the radius of the inscribed circle of triangle $A B C$.

## Hint

Prove that the midpoint of segment $P Q$ is the center of the inscribed circle of triangle $A B C$.

## Solution

Let $D, E, F$ be the feet of the perpendiculars dropped from points $P, Q, O$ to the line $A B$ respectively. By the problem's condition, $P D=6, Q E=10$. $O F$ is the midline of trapezoid $D P Q E$, so the length of $O F$ is $1 / 2(P D+Q E)=1 / 2(6+10)=8$. Therefore, point $O$ is at a distance of 8 from the line $A B$. Similarly, we find that $O$ is at a distance of $1 / 2(7+9)=8$ from the line $B C$ and at a distance of $1 / 2(12+4)=8$ from the line $C A$. Thus, $O$ is the center of the inscribed circle of triangle $A B C$, and the radius of this circle is 8.

## Answer

8.