CAPS 2014

Prove that if the positive real numbers $a, b, c$ satisfy the equation $$a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),$$ then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that $$\sin \alpha = a, \sin \beta = b, \sin \gamma= c.$$

For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that $x_n = ax_{n-1} + b$ for each $n \ge 2$ . Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$ .

(Jaromír Šimša)

Given is a convex $ABCD$ , which is $|\angle ABC| = |\angle ADC|= 135^\circ$ . On the $AB, AD$ are also selected points $M, N$ such that $|\angle MCD| = |\angle NCB| = 90^ \circ$ . The circumcircles of the triangles $AMN$ and $ABD$ intersect for the second time at point $K \ne A$ . Prove that lines $AK$ and $KC$ are perpendicular.

(Irán)

Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$ . A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$ . Let $S$ be the midpoint of $KL$ . Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$ , then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$ .

(Jan Mazák)

Let all positive integers $n$ satisfy the following condition: for each non-negative integers $k, m$ with $k + m \le n$ , the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$ .

(Poland)

PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

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Let $n \ge 6$ be an integer and $F$ be the system of the $3$ -element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition: for every $1 \le i < j \le n$ there is at least $\lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$ . Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F$ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5$ (Poland)

PS. just in case my translation does not make sense, I leave the original in Slovak, in case someone understands something else

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