MYTS 2017 - Grade 10 - Round 1

Solve the equation √(x+17)+√x=17.

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

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 greatest value of f(x)=sin⁶x+cos⁶x+3 sin²x cos²x.

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?

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

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 function f:ℝ→ℝ is nondecreasing: $a<b$ implies $f(a)≤f(b)$. Also f(1)=1. How many real x satisfy f(1+x)=1−x?

The two distinct parabolas y=x²+ax+b and y=x²+bx+a intersect the axes at A,B,C,D,E as shown and meet at B on the horizontal axis. A=(−8,0). Find the area of quadrilateral ADCE.

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.

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?

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.

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.

Real numbers a,b,c,d satisfy a+b+c+d=a²+b²+c²+d². Let T be the least value of (a+b)(c+d)+ab+cd. Find 100+8 T.

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

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

February 2012 had exactly five Wednesdays. What is the next year whose February also has exactly five Wednesdays?

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