2019 Hong Kong TST

August 18, 2018 - Test 1

1

Let $a$ be a real number. Suppose the function $f(x) = \frac{a}{x-1} + \frac{1}{x-2} + \frac{1}{x-6}$ defined in the interval $3 < x < 5$ attains its maximum at $x=4$. Find the value of $a.$

2

A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.

3

Find an integral solution of the equation \[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \](Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$.)

4

Let $ABC$ be an acute-angled triangle such that $\angle{ACB} = 45^{\circ}$. Let $G$ be the point of intersection of the three medians and let $O$ be the circumcentre. Suppose $OG=1$ and $OG \parallel BC$. Determine the length of the segment $BC$.

5

Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

6

If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of \[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]

October 20, 2018 - Test 2

1

Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation \[ p_{n+2} = p_{n+1} + p_n + k \]holds for all positive integers $n$.

2

Let $p$ be a prime number greater than 10. Prove that there exist positive integers $m$ and $n$ such that $m+n < p$ and $5^m 7^n-1$ is divisible by $p$.

3

Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$. (a) Prove that $\angle{EPF} = 2\angle{CAE}$. (b) Prove that $AP^2 = CA^2 + PE^2$.

4

We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.

The CHKMO and APMO are used as selection tests in between these Test 2 and 3. - Note

April 28, 2019 - Test 3

1

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$for all $x,y\in\mathbb{Q}_{>0}$

2

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

3

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

May 1, 2019 - Test 4

1

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

3

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. Proposed by Evan Chen, Taiwan