Problem 1 Find all quadruples of real numbers such that the equalities hold for all real numbers .
Problem 2 The Bank of Zürich issues coins with an on one side and a on the other side. Alice has of these coins arranged in a line from left to right. She repeatedly performs the following
operation: if some coin is showing its side, Alice chooses a group of consecutive coins
(this group must contain at least one coin) and flips all of them; otherwise, all coins show and Alice stops. For instance, if , Alice may perform the following operations: . She might also choose to perform the
operation .
For each initial configuration , let be the minimal number of operations that
Alice must perform. For example, and . For every integer , determine the largest value of over all 2
n possible initial configurations .
Problem 3 Let and be two distinct points in the plane. Let be the midpoint of the segment $AB$, and let $\omega$ be a circle that goes through $A$ and $M$. Let $T$ be a point on $\omega$ such that
the line $BT$ is tangent to $\omega$. Let $X$ be a point (other than $B$) on the line $AB$ such that $TB = TX$, and let $Y$ be the foot of the perpendicular from $A$ onto the line $BT$.
Prove that the lines $AT$ and $XY$ are parallel.
Problem 4 For all real numbers $x$, we denote by $\lfloor x\rfloor$ the largest integer that does not exceed $x$. Find
all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy
the equality for all real numbers $x$ and $y$
Problem 5 Let $n$ and $k$ be positive integers such that $k \leq 2n$ . Banana and Corona are playing the
following variant of the guessing game. First, Banana secretly picks an integer $x$ such
that $1 \leq x \leq n$. Corona will attempt to determine $x$ by asking some questions, which
are described as follows. In each turn, Corona chooses $k$ distinct subsets of $\{1, 2, \cdots , n\}$ and, for each chosen set $S$, asks the questionBanana picks one of these $k$ questions and tells both the question and its answer to
Corona, who can then start another turn.
Find all pairs $(n, k)$ such that, regardless of Banana’s actions, Corona could determine $x$ in finitely many turns with absolute certainty.
Problem 6 For every integer $n$ not equal to 1 or −1, define $S(n)$ as the smallest integer greater than $1$ that divides $n$. In particular, $S(0) = 2$. We also define $S(1) = S(−1) = 1$.
Let $f$ be a non-constant polynomial with integer coefficients such that $S(f(n)) \leq S(n)$ for every positive integer $n$. Prove that $f(0) = 0$.
Note: A non-constant polynomial with integer coefficients is a function of the form $f(x) = a_0 + a_1x + a_2x^2 + \cdots + a_kx^k$ , where $k$ is a positive integer and $a_0$, $a_1$,$\cdots$ , $a_k$ are
integers such that $a_k \neq 0$.
Advanced Level:-
Day 1:-
Time: 5 Hours
Each problem is worth 7 points
Problem 1 Let $ABC$ be a triangle with incentre $I$. The incircle of the triangle $ABC$ touches the sides $AC$ and $AB$ at points $E$ and $F$, respectively. Let $l_B$ and $l_C$ be the tangents to the circumcircle of $BIC$ at $B$ and $C$, respectively. Show that there is a circle tangent to $EF$, $l_B$ and $l_C$ with centre on the line $BC$.
Problem 2 Geoff has an infinite stock of sweets, which come in $n$ flavours. He arbitrarily distributes some of
the sweets amongst $n$ children (a child can get sweets of any subset of all flavours, including the
empty set). Call a distribution of sweets $k-$nice if every group of $k$ children together has sweets in
at least $k$ flavours. Find all subsets $S$ of $\{1, 2,\cdots , n\}$ such that if a distribution of sweets is $s-$nice
for all $s \in S$, then it is $s-$nice for all $s \in \{1, 2,\cdots , n\}$.
Problem 3 We call a set of integers special if it has 4 elements and can be partitioned into 2 disjoint subsets $\{a, b\}$ and $\{c, d\}$ such that $ab−cd = 1$. For every positive integer $n$, prove that the set $\{1, 2,\cdots , 4n\}$ cannot be partitioned into $n$ disjoint special sets.
Problem 4 Prove that, for all sufficiently large integers n, there exist n numbers $a_1, a_2,\cdots , a_n$ satisfying the
following three conditions:
• Each number $a_i$ is equal to either −1, 0 or 1.
• At least $\frac{2n}{5}$ of the numbers $a_1, a_2,\cdots , a_n$ are non-zero.
• The sum $\frac{a_1}{1} + \frac{a_2}{2} + \cdots + \frac{a_n}{n}$ is 0.
Note: Results with $\frac{2}{5}$ replaced by a constant $c$ will be awarded points depending on the value of $c$.
Day 2:-
Time: 5 Hours
Each problem is worth 7 points
Problem 5 Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$
Problem 6 Decide whether there exist infinitely many triples $(a, b, c)$ of positive integers such that all prime
factors of $a! + b! + c!$ are smaller than 2020.
Problem 7 Each integer in $\{1, 2, 3, \cdots , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$ such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
Problem 8 Let $ABC$ be an acute scalene triangle, with the feet of $A$, $B$, $C$ onto $BC$, $CA$, $AB$ being $D$, $E$, $F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC$, $CA$, $AB$ are $W_a$, $W_b$, $W_c$ respectively. Finally, let $N$ and $I$ be the circumcentre and incentre of $W_aW_bW_c$ respectively. Prove
that, if $N$ coincides with the nine-point centre of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.
Note: If $XYZ$ is a triangle with circumcentre $O$ and orthocentre $H$, then the line $OH$ is called the
Euler line of $XYZ$ and the midpoint of $OH$ is called the nine-point centre of $XYZ$
You can download the questions from here:-
