Beginner Level:-
Time: 5 Hours
Each problem is worth 7 points
Each problem is worth 7 points
Problem 1 Find all quadruples of real numbers \((a, b, c, d)\) such that the equalities $$X^2 + aX + b = (X − a)(X − c) \text{ and }\ X^2 + cX + d = (X − b)(X − d)$$hold for all real numbers \(X\).
Problem 2 The Bank of Zürich issues coins with an \(H\) on one side and a \(T\)on the other side. Alice has \(n\) of these coins arranged in a line from left to right. She repeatedly performs the following operation: if some coin is showing its \(H\) 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 \(T\) and Alice stops. For instance, if \(n = 3\), Alice may perform the following operations: \(THT \to HTH \to HHH \to TTH \to TTT\). She might also choose to perform the operation \(THT \to TTT\).
For each initial configuration \(C\), let \(m(C)\) be the minimal number of operations that Alice must perform. For example, \(m(THT) = 1\) and \(m(TTT) = 0\). For every integer \(n\geq 1\), determine the largest value of \(m(C)\) over all 2 n possible initial configurations \(C\).
Problem 3 Let \(A\) and \(B\) be two distinct points in the plane. Let \(M\) 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 $$f(x + y) = (−1)^{\lfloor x\rfloor} f(x) + (−1)^{\lfloor x\rfloor} f(y)$$ 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 question$$“\text{Is}\ x\ \text{in the set}\ S?”$$Banana 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 $a_k \neq 0$.
Problem 2 The Bank of Zürich issues coins with an \(H\) on one side and a \(T\)on the other side. Alice has \(n\) of these coins arranged in a line from left to right. She repeatedly performs the following operation: if some coin is showing its \(H\) 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 \(T\) and Alice stops. For instance, if \(n = 3\), Alice may perform the following operations: \(THT \to HTH \to HHH \to TTH \to TTT\). She might also choose to perform the operation \(THT \to TTT\).
For each initial configuration \(C\), let \(m(C)\) be the minimal number of operations that Alice must perform. For example, \(m(THT) = 1\) and \(m(TTT) = 0\). For every integer \(n\geq 1\), determine the largest value of \(m(C)\) over all 2 n possible initial configurations \(C\).
Problem 3 Let \(A\) and \(B\) be two distinct points in the plane. Let \(M\) 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 $$f(x + y) = (−1)^{\lfloor x\rfloor} f(x) + (−1)^{\lfloor x\rfloor} f(y)$$ 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 question$$“\text{Is}\ x\ \text{in the set}\ S?”$$Banana 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
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
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 $x, y \in \mathbb{Q}$ $$f(x)f(y + 1) = f(xf(y)) + f(x)$$
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 $XYZ$
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:-
GQMO 2020: Beginner Level
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