MYTS 2018 — Grade 8 — Round 1

Positive numbers $x,y$ satisfy $x>y$ and $(x-2y)^2/(xy)=8/3$. Find $x/y$.

Let $A=2018^2-2017^2+2016^2-2015^2+\cdots+2^2-1^2$. Find its last two digits.

Ánh Viên’s 200 m freestyle time at SEA Games 28 is 1 minute 59 seconds. After the starting dive she swims steadily at 1.6 m/s. Ignoring the dive’s duration, how far does the starting dive carry her, in metres?

How many positive integers $n$ have remainder equal to the square of the quotient when divided by 45?

How many integer pairs $(x,y)$ satisfy $|x|+|y|\le7$?

For $x,y>0$, let $a=x+1/x$, $b=y+1/y$, $c=xy+1/(xy)$. Simplify $P=a^2+b^2+c^2-abc$.

Lan has 37 sweets, Thảo 49 and Mai 67. Each day one of them gives one sweet to every other classmate, excluding these three. When no further such distribution is possible, all three have equal remainders. What is the largest possible class size?

Xuân Trường and Văn Thanh practise penalties, each taking at least one shot. Their exact success rates are 80% and 95%, respectively. With 125 shots in total, what is the greatest combined number of successful shots?

$E$ is the midpoint of side $CD$ of square $ABCD$, whose area is 144 cm². Find the shaded area in cm².

Farms $A,B,C,D$ lie in order on a straight road, with $AB=2$ km, $AC=5$ km and $AD=6$ km. Place a factory on the road so its distances to $A,B$ sum to its distances to $C,D$. How far is it from $A$, in km?

For convex hexagon $ABCDEF$, how many vertex colourings use red, blue, yellow exactly twice each, with the two vertices of each colour nonadjacent? Colourings are different if some labelled vertex has a different colour.

A rectangular tank has a square base of side 1 m and height 2 m. A bottom valve drains 0.1 m³/h; another valve 0.6 m above the bottom drains 0.05 m³/h. Water enters at 0.2 m³/h. The tank starts empty and both valves stay open. After how many hours does it begin to overflow?

How many integer triples $(a,b,c)$ satisfy $ab+bc+ca=0$ and $1/a^3+1/b^3+1/c^3+3/4=0$?

A class has 8 students in rowI and 10 in rowII, with average maths scores 9 and 8.1. If Xuyến from rowI swaps places with Diệp from rowII, the row averages become equal. How much do their scores differ?

Triangle $ABC$ has $AB=AC=17$, $BC=30$. Construct triangle $BCD$ outside it with $\angle CBD=90^\circ$ and $CD\parallel AB$. Find $BD$.

Find the smallest positive multiple of 99 whose decimal representation contains no digit 9.

Find the smallest three-digit prime $\overline{abc}$ for which $a,\overline{ab},\overline{abc},c,\overline{cb},\overline{cba}$ are all prime.

Real numbers $a,b,c$ satisfy $|a|\le5$, $|b+1|\le7$, $|c+2|\le9$. Find the minimum $S$ and maximum $T$ of $|a+b-c|$.

Real numbers $a,b,c,d$ satisfy $(a+b)(c+d)=2$, $(a+c)(b+d)=3$, $(a+d)(b+c)=4$. Find the minimum of $a^2+b^2+c^2+d^2$.

Thirteen circles form three rows of five as shown. Place the numbers 1 through 13 once each so that every row of five sums to the sum of the seven shaded circles. What is the greatest sum of the six numbers in the ×-marked circles?