IL-GILLIS 2021

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints).a) Which digit did Sophie write the most?b) Which digit did Sophie write the least?

Does there exist an infinite sequence of primes $p_1, p_2, p_3, \dots$ for which, $$p_{n+1}=2p_n+1$$ for each $n$ ?

Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$ . Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that $$AX+AY+BC>AB+AC$$

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Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$ .

Is the amount of numbers Danny likes divisible by $7$ ?

21 players participated in a tennis tournament, in which each pair of players played exactly once and each game had a winner (no ties are allowed). The organizers of the tournament found out that each player won at least 9 games, and lost at least 9. In addition, they discovered cases of three players $A,B,C$ in which $A$ won against $B$ , $B$ won against $C$ and $C$ won against $A$ , and called such triples "problematic".a) What is the maximum possible number of problematic triples?b) What is the minimum possible number of problematic triples?

Triangle $ABC$ is given. The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively. When $ABC$ is rotated $180$ degrees about point $I$ , triangle $A'B'C'$ results. Lines $AD, B'C'$ meet at $U$ , lines $BE, A'C'$ meet at $V$ , and lines $CF, A'B'$ meet at $W$ . Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively. Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively. Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively. Six (not necessarily convex) quadrilaterals were colored orange: $$AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2$$ Six other quadrilaterals were colored green: $$AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1$$ Prove that the sum of the green areas equals the sum of the orange areas.

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