MYTS 2018 - Grade 6 - Round 2

Seven distinct collinear points occur from left to right as A,B,C,D,E,F,H. C is the midpoint of AE, and D is the midpoint of each of AH,BF,CE. Find AC/AH.

For natural number N let S(N) be its digit sum. Find the smallest N for which both S(N) and S(N+1) are divisible by 7.

Five rays Oa,Ob,Oc,Om,On are ordered as in Figure 1. Om bisects an angle made by two of the other four rays. The angles aOb,bOc,cOm,mOn strictly increase in that order. Three of these four angles are 10°,15°,30°. Find all possible values of angle mOn.

Find all four-digit numbers abcd for which gcd(abcd,ab)=8(a+b+c+d), where ab denotes the two-digit prefix.

Five people sit around a circular table, on chairs numbered 1–5 clockwise. One stole a golden spoon. Their statements are:1: “The thief is immediately to my left.”2: “The thief is on an odd-numbered chair.”3: “The thief sits next to me.”4: “The thief is immediately to my left.”5: “I did not steal it.” Exactly three statements are true. Which chair holds the thief?

Lan wants to tile an 8 × 9 rectangle (Figure 2) without overlap, using 2 × 2 squares (Figure 3) and 6 × 1 rectangles (Figure 4). (a) Is tiling possible using only 2 × 2 squares? Explain. (b) Is it possible using only 6 × 1 rectangles? Explain.