MYTS 2017 - Grade 7 - Round 1

The graph y=|x| intersects y=10 at (a,b) and (c,d). Find a+b+c+d.

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.

In triangle ABC, the internal bisectors of angles ABC and ACB meet at D. If angle BDC=114°, find angle CAB in degrees.

Positive integers a,b satisfy a^b b^a=72. What is the greatest possible value of a+b?

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?

ABC is isosceles at A, with AB=AC=7 cm and BC=2 cm. Point D lies on ray BC and AD=8 cm. Find BD in cm.

Real numbers x,y satisfy x/3=y/5 and xy²=600. Find |10 x−3 y|.

Hạnh compares a phone costing USD 120 plus USD 30 per month with the same kind of phone costing USD 40 plus USD 40 per month. After how many months are the total costs equal?

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

Four unit circles are described in the source as “pairwise tangent”, arranged as shown. Each side of the rectangle is tangent to exactly one circle. Its length is ℓ=c+a√b, where a,b,c are positive integers and b is not a perfect square. Find a+b+c.

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

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.

ABC is right-angled at A. Points D,E lie on AC, and angle ACB=angle CBE=angle ABD=19°. If AE/CD=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.

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

Six typists working 5 hours per day complete a manuscript in 16 days. How many days do four equally productive typists need if each works 6 hours per day?

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

Find the sum of the nine marked angles A+B+C+⋯+H+I in the figure, in degrees.