## SUBJECT II

On the circles in the figure are placed the numbers $3, 7, 11, 15$,

a) Write the numbers that will be placed on the next two circles;

b) Determine the number that must be placed on the 2015th circle;

c) Show that the number on the 2015th circle is not a perfect square;

d) Calculate the sum of the numbers on the first 2015 circles.

![](https://cdn.mathpix.com/cropped/2024_06_07_cc8ac07229c0dd5d256fg-2.jpg?height=525&width=489&top_left_y=1762&top_left_x=1566)
## Solution:

a) 19.23

b) The numbers on the circles are of the form $4k+3$, so on the 2015th circle it is $4 \cdot 2014 + 3 = 8059$ $2p$

c) The number on the 2015th circle is between two consecutive squares $89^2 < 8059 < 90^2$

Or: numbers of the form $4k+3$ are not perfect squares.

![](https://cdn.mathpix.com/cropped/2024_06_07_cc8ac07229c0dd5d256fg-3.jpg?height=259&width=294&top_left_y=119&top_left_x=356)
d) $S = 4 \cdot 0 + 3 + 4 \cdot 1 + 3 + 4 \cdot 2 + 3 + \ldots$ $\qquad$ $+ 4 \cdot 2014 + 3 = 4 \cdot (0 + 1 + 2 + 3 + \ldots + 2014) + 3 \cdot 2015 = 2014 \cdot 2015 \cdot 2 + 3 \cdot 2015 = 2015 \cdot 4031 = 8122465$ $2p$