MYTS 2018 — Grade 10 — Round 1

Simplify $P=\sqrt{(\sin^8x+\cos^8x+1)/2}$.

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

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

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 $T=a+b+c$.

Let $f:(0,+\infty)\to(0,+\infty)$ satisfy $xf(x)-yf(y)=f(y)-f(x)$ for all positive $x,y$. Given $f(2018)=2$, find $f(1)$.

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 $c$.

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

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

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

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 ×-marked circles. What is the greatest sum of the six numbers in the shaded circles?

Let $f:\mathbb R\to\mathbb R$ satisfy $f(-x)=f(x)=f(x+6)$ for every real $x$, and $f(x)=3/x$ for $0<x\le3$. Find $f(2018)$.

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?

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 $CJ$.

Find all real $a$ such that the real roots $x_1,x_2$ of $x^2+ax-1=0$ satisfy $1/(x_1-1)^2+1/(x_2-1)^2=7$.

In square $ABCD$, points $E,F$ lie on sides $AB,BC$ so that $DE,DF$ divide the square into three equal-area parts. Find $\tan\angle EDF$.

Let $f(x)=x^3-33x^2+363x-1320$. For integers $n\ge2$, define $f_n(x)=f(f_{n-1}(x))$, with $f_1=f$. Solve $f_{2018}(x)=2007^{3^{2018}}+11$.

In the figure, $AB=BC=CD=6$, $PB=5$, $PC=7$. The two circles intersect at $P,Q$ as shown. Find $PQ$.

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.