IrMO 2019

Define the quasi-primes as follows. $\bullet$ The first quasi-prime is $q_1 = 2$ $\bullet$ For $n \ge 2$ , the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$ . Determine, with proof, whether or not $1000$ is a quasi-prime.

Jenny is going to attend a sports camp for $7$ days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of $4$ consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.

A quadrilateral $ABCD$ is such that the sides $AB$ and $DC$ are parallel, and $|BC| =|AB| + |CD|$ . Prove that the angle bisectors of the angles $\angle ABC$ and $\angle BCD$ intersect at right angles on the side $AD$ .

Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy $$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$

Let $M$ be a point on the side $BC$ of triangle $ABC$ and let $P$ and $Q$ denote the circumcentres of triangles $ABM$ and $ACM$ respectively. Let $L$ denote the point of intersection of the extended lines $BP$ and $CQ$ and let $K$ denote the reflection of $L$ through the line $PQ$ . Prove that $M, P, Q$ and $K$ all lie on the same circle.

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises both (a) and (b). (You may assume that (1) and (2) are correct!)

Three non-zero real numbers $a, b, c$ satisfy $a + b + c = 0$ and $a^4 + b^4 + c^4 = 128$ . Determine all possible values of $ab + bc + ca$ .

Consider a point $G$ in the interior of a parallelogram $ABCD$ . A circle $\Gamma$ through $A$ and $G$ intersects the sides $AB$ and $AD$ for the second time at the points $E$ and $F$ respectively. The line $FG$ extended intersects the side $BC$ at $H$ and the line $EG$ extended intersects the side $CD$ at $I$ . The circumcircle of triangle $HGI$ intersects the circle $\Gamma$ for the second time at $M \ne G$ . Prove that $M$ lies on the diagonal $AC$ .

Suppose $x, y, z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$ . Prove that $8xyz \le 1$ , with equality if and only if $(x, y,z)$ is one of the following: $$\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)$$

Island Hopping Holidays offer short holidays to $64$ islands, labeled Island $i, 1 \le i \le 64$ . A guest chooses any Island $a$ for the first night of the holiday, moves to Island $b$ for the second night, and finally moves to Island $c$ for the third night. Due to the limited number of boats, we must have $b \in T_a$ and $c \in T_b$ , where the sets $T_i$ are chosen so that (a) each $T_i$ is non-empty, and $i \notin T_i$ , (b) $\sum^{64}_{i=1} |T_i| = 128$ , where $|T_i|$ is the number of elements of $T_i$ . Exhibit a choice of sets $T_i$ giving at least $63\cdot 64 + 6 = 4038$ possible holidays. Note that c = a is allowed, and holiday choices $(a, b, c)$ and $(a',b',c')$ are considered distinct if $a \ne a'$ or $b \ne b'$ or $c \ne c'$ .