MYTS 2018 — Grade 9 — Round 1

Solve $\sqrt{2x+1}=x-3/2$.

A positive integer $n$ leaves remainder 1 modulo 3. Find the remainder of $n(n+1)/2$ modulo 9.

Square $ABCD$ has side 1. Arcs of radius 1 are drawn with centres $A,B,C$ as shown. Find the length of arc $MN$.

The four roots of $P(x)=x^4+ax^3+bx^2+cx+1$ are the side lengths of a parallelogram. Find the minimum of $T=a+b+c$.

Find the smallest perfect square divisible by 11 and leaving remainder 1 modulo 7.

Trapezoid $ABCD$ has $AB\parallel CD$, $AB=BC=17$, $AC=30$ and $\angle CAD=90^\circ$. Find $AD$.

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 convex polygon with 2018 vertices is triangulated by diagonals not crossing in its interior. Let $a,b,c$ count triangles with 2,1,0 polygon sides, respectively. Given $a=504b$, find $c$.

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 $a^2+b^2+c^2+d^2$.

Primes $p,q,r$ satisfy $(p+q)/r=p-q+r$ and $p+q<120$. Find the maximum of $pqr$.

In the figure, $ABC$ is equilateral, $D$ is the midpoint of $AC$, and $DE\parallel BC$, with $E$ on the circumcircle as shown. Find $DE/BC$.

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?

The parallel lines $d_1,d_2,d_3$ are 1 cm and 3 cm apart consecutively, as shown. Equilateral triangle $ABC$ has its vertices on the three lines. Find its side length in cm.

There are 21 bottles: seven containing 1 litre, seven containing $2/3$ litre and seven containing $1/3$ litre. Divide the intact bottles among 3 people so that both bottle counts and milk volumes are equal. How many bottles of each type does each person receive, in the order $1,2/3,1/3$ litre?

Positive integers $a,b<1000$ have arithmetic and geometric means that are consecutive odd integers. Find the maximum of $|a-b|$.

Square $ABCD$ has side 1. Point $E$ lies on $CD$, excluding endpoints $C,D$. Construct square $AEGF$ as shown. Let $J=BC\cap EG$. Find the maximum length $CJ$.

Solve $\lfloor x\rfloor\{x\}=2018x$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part.

Let $n$ be a positive integer below 10. Concatenate the digits 1 through $n$ in any order ending with $n$, to obtain $N$. Let $S(N)$ be its digit sum. Find all $n$ for which $S(N)-N$ is divisible by 45.

Square $ABCD$ has vertices $A,B$ on the circle centred at $O$ of radius 1, as shown. A circle centred at $O_1$ with radius $r$ is tangent to $(O)$, segment $PD$ and $AD$. Find $r$.

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?