p1. Find all the primes $p$ such that $p^2 + 2$ is a prime number. p2. In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest of the circles has radius $8$ and the largest has radius $18$. Calculate the radius of the circle $C$. p3. Consider a network composed of four regular hexagons as a sample in the figure: A bee and a fly play the following game: initially the bee chooses one of the dots and paints it red, then the fly chooses one of the unpainted dots and paints it blue. Then the bee chooses an unpainted spot and paints it red and then the fly chooses an unpainted one and paints it blue and so they alternate. If in the end of the game there is an equilateral triangle with its red vertices, the bee wins, otherwise the fly wins. Determine which of the two insects has a winning strategy. p4. A box contains $15$ red pencils, $13$ blue pencils, and $8$ green pencils. Someone asks Constanza to take a number of pencils out of the box blindfolded. What is the minimum number of pencils that Constanza should take out in order to make sure she gets at least $ 1$ red, $2$ blue, and $3$ green? p5. Let us call a $ 12$-sided regular polygon $P$. How many triangles is it possible to form using the vertices of $P$? How many of them are scalene triangles? p6. Consider two lines $L_1, L_2$ that are cut at point $O$ and $M$ is the bisector of the angle they form, as shown in the following figure. Points $A$ and $B$ are drawn in $M$ in such a way that $OA = 8$ and $OB = 15$ and the angle $\angle L_1OL_2$ measures $45^o$ . Calculate the shortest possible path length from $A$ to $B$ by touching lines $L_1$ and $L_2$.
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Tags: algebra, geometry, combinatorics, number theory, chilean NMO
BackToSchool
21.10.2021 16:23
Obviously, $p=3$ is a solution.
For prime $p \ge 5$, it can be written as $p=6k\pm1$, thus $p^2 + 2=36k^2\pm12k+3$ is multiple of $3$.
Therefore, $p=3$ is the only solution.
BackToSchool
21.10.2021 16:35
We need to consider the worst case: Constanza takes out 21 pencils, of which 13 are blue pencils and 8 are green pencils.
She just need to take another one to make sure she gets at least $ 1$ red, $2$ blue, and $3$ green pencils.
Therefore, she should take out minimum $\boxed {22}$ pencils.
JanHaj
27.06.2023 23:23
p1 First let's check the cases when: $p=2 \Rightarrow p^2+2=4+2=6$ which obviously isn't prime $p=3 \Rightarrow p^2+2=9+2=11$ so $p=3$ is a solution. Now we will show that $p>3$ is not possible, we have: Case 1:$p\equiv 1\pmod{3}$ $\Rightarrow p^2+2\equiv 1+2\equiv0\pmod{3}$ Case 2:$p\equiv 2\pmod{3}$ $\Rightarrow p^2+2\equiv 6\equiv0\pmod{3}$ In both cases we have that $p^2+2\equiv0\pmod{3}$ which we can't have. Thus, $p=3$ is the only solution.