MYTS 2017 - Grade 8 - Round 1

Given x²−3 x+1=0, find x⁴+1/x⁴.

A rectangular tank has length, width and height in ratio 1:2:4. Its total six-face surface area is 112 m². Find its volume in m³.

The picture gives two flowers plus one can equal 32, and two cans plus one flower equal 43. What is one flower plus one can?

The curves are the graph y=a/x. It passes through K(1/6,2) and a point (b,c) to the right of the vertical axis, with b−48 c=0. Find b+120 c.

Find the smallest seven-digit natural number with distinct digits, divisible by 7, whose digit sum is also divisible by 7.

Two semicircles with diameters AB and DC are drawn inside rectangle ABCD and intersect at two points. AB=104 cm and BC=96 cm. Find the distance between the intersection points in cm.

Real numbers x,y,z satisfy x/2=y/3=z/4 and |x−y|=z²/12. Find the greatest value of yz−x.

Find the remainder when 2^2017 is divided by 37.

An equilateral triangle, a square and a regular pentagon share vertices as shown. AK and HL are perpendicular to KL. Using the marked 21°,55°,39°, find the marked angle at E in degrees.

Label the eight vertices of a cube with 1,2,…,8, once each, to maximize the number of edges whose endpoint labels have odd sum. What is that maximum?

Three meshing gears A,B,C have 16,22,10 teeth respectively. A turns at 60 revolutions per minute. How many revolutions per minute does C make?

Choose 7 distinct points on line k and 9 distinct points on a parallel line ℓ. How many triangles have all three vertices among these 16 points?

Find the last two digits of A=(1+2+⋯+2016+2017)².

ABCD is a rhombus. E lies on AB and F on CD, with CE=CA and CF=DF. If AF/DE=m/n in lowest positive terms, find 3 m+5 n.

Find the smallest three-digit natural number N divisible by 77 whose digits are all odd.

ABCDEF is a regular hexagon. G lies on AF with GF=(2/5)AF. Its area is 120 cm². Find the shaded area in cm².

What is the greatest possible number of primes among 100 consecutive natural numbers?

An integer n≥2 satisfies n²≡1 modulo 10. Find the remainder of n² upon division by 20.

A regular polygon is divided into 18 congruent pentagons as shown; all sides of each pentagon are equal. If k,ℓ are the two largest angles of one pentagon, find k+ℓ in degrees.

Nine cards carry the digits 1 through 9. How many ways are there to choose seven cards whose digit sum is divisible by 3?