MYTS 2017 - Grade 9 - Round 1

Solve the equation √(x+17)+√x=17.

In the Oxy plane, y=a−x²/b intersects the horizontal axis at (8,0),(−8,0) and the vertical axis at (0,8). Find a+b.

Given (√a+1/2)²=1 and (√b−1/2)²=1, find √b−√a.

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?

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

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.

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

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

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

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

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.

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?

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?

ABCD is circumscribed about circle (I,r), with CD=5. Circles (K),(L) have equal radius 2 and KL=3, with the tangencies and centre order shown. Find r.

How many ordered pairs of positive integers (x,y) satisfy 1/x+1/y=1/2020?

Find the smallest natural number N divisible by 99 whose digits are all even.