A magician has $300$ cards with numbers from $1$ to $300$ written on them, each number on exactly one card. The magician then lays these cards on a $3 \times 100$ rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over $k$ cards of his choice. What is the smallest value of $k$ for which it can happen that the opened cards definitely determine the exact positions of all other cards?

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

In a new edition of QoTD duels, $n \ge 2$ ranked contestants (numbered 1 to $n$) play a round robin tournament (i.e. each pair of contestants compete against each other exactly once); no draws are possible. Define an upset to be a pair $(i, j)$ where$ i > j$ and contestant $i$ wins against contestant $j$. At the end of the tournament, contestant $i$ has $s_i$ wins for each $1 \le i \le n$. The result of the tournament is defined as the $n$-tuple $(s_1, s_2, \cdots , s_n)$. An $n$-tuple $S$ is called interesting if, among the distinct tournaments that produce $S$ as a result, the number of tournaments with an odd number of upsets is not equal to the number of tournaments with an even number of upsets. Find the number of interesting $n$-tuples in terms of $n$. [i](Two tournaments are considered distinct if the outcome of some match differs.)[/i]

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(f(x+1)) = x^3+1$ for all real numbers $x$. Prove that the equation $f(x) = 0 $ has exactly one real root.

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Suppose that $k$ is a positive integer. A bijective map $f : Z \to Z$ is said to be $k$-[i]jumpy [/i] if $|f(z) - z| \le k$ for all integers $z$. Is it that case that for every $k$, each $k$-jumpy map is a composition of $1$-jumpy maps? [i]It is well known that this is the case when the support of the map is finite.[/i]

Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ). Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$. [i](No instruments are allowed, even a pencil.)[/i] (E. Bakayev, A. Zaslavsky)

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$. (Tran Quang Hung)

A partition of a set \( A \) is a family of non-empty subsets of \( A \), such that any two distinct subsets in the family are disjoint, and the union of all subsets equals \( A \). We say that a partition of a set of integers \( B \) is [i]separated[/i] if each subset in the partition does [b]not[/b] contain consecutive integers. Prove that, for every positive integer \( n \), the number of partitions of the set \( \{1, 2, \dots, n\} \) is equal to the number of separated partitions of the set \( \{1, 2, \dots, n+1\} \). For example, \( \{\{1,3\}, \{2\}\} \) is a separated partition of the set \( \{1,2,3\} \). On the other hand, \( \{\{1,2\}, \{3\}\} \) is a partition of the same set, but it is not separated since \( \{1,2\} \) contains consecutive integers.

Let $f:\mathbb{N}\to\mathbb{N}$ be a bijection of the positive integers. Prove that at least one of the following limits is true: \[\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{n+f(n)}=\infty;\qquad\lim_{N\to\infty}\sum_{n=1}^N\left(\frac{1}{n}-\frac{1}{f(n)}\right)=\infty.\][i]Proposed by Dylan Toh[/i]

Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?

Let $ABCD$ be a convex quadrilateral and $K$ be the common point of rays $AB$ and $DC$. There exists a point $P$ on the bisectrix of angle $AKD$ such that lines $BP$ and $CP$ bisect segments $AC$ and $BD$ respectively. Prove that $AB = CD$.

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$. (F.Ivlev)

For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$. (i): $n=a_1+a_2+\cdots +a_l$ (ii): $a_1>a_2>\cdots > a_l > 0$. (iii): $a_l$ is an odd number. (iv): There are $k$ odd numbers out of $a_i$. For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$. Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that: [i]a)[/i] all of the good lines in a triangle concur. [i]b)[/i] all of the good lines in a regular polygon concur too.

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]