2. In triangle $\triangle A B C$, the sides $A B=5$ and $A C=6$ are known. What should the side $B C$ be so that the angle $\angle A C B$ is as large as possible? Provide the length of side $B C$, rounded to the nearest integer.
Answer: 3

Solution: Construct $AC=6$. Then the geometric locus of points $B$ will be a circle of radius 5 centered at point $A$. The angle $\angle ACB$ will be the largest when $CB$ is tangent to the circle (see figure). Then $CB \perp AB$ and by the Pythagorean theorem we get $BC=\sqrt{AC^{2}-AB^{2}}=\sqrt{11} \approx 3$.

## 2013/2014 Academic Year CRITERIA FOR DETERMINING WINNERS AND PRIZE WINNERS ${ }^{1}$

## of the school students' competition "CONQUER SPARROW MOUNTAINS!" IN MATHEMATICS

ELIMINATION STAGE

WINNER:

From 95 points inclusive and above.

PRIZE WINNER:

From

91 points to

94

points inclusive.

FINAL STAGE

WINNER (Diploma I degree):

From 90 points inclusive and above.

PRIZE WINNER (Diploma II degree):

From 75 points to 89 points inclusive.

PRIZE WINNER (Diploma III degree):

from 60 points to 74 points inclusive.[^0]


[^0]:    ${ }^{1}$ Approved at the meeting of the jury of the school students' competition "Conquer Sparrow Mountains!" in mathematics