Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that \[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]

For a positive integer $n$, let $\mathcal{D}(n)$ be the value obtained by, starting from the left, alternating between adding and subtracting the digits of $n$. For example, $\mathcal{D}(321)=3-2+1=2$, while $\mathcal{D}(40)=4-0=4$. Compute the value of the sum \[\sum_{n=1}^{100}\mathcal{D}(n)=\mathcal{D}(1)+\mathcal{D}(2)+\dots+\mathcal{D}(100).\]

Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

Source: 2018 Canadian Open Math Challenge Part A Problem 1 ----- Suppose $x$ is a real number such that $x(x+3)=154.$ Determine the value of $(x+1)(x+2)$.

A class with $2N$ students took a quiz, on which the possible scores were $0,1,\dots,10.$ Each of these scores occurred at least once, and the average score was exactly $7.4.$ Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4.$

Given a positive integer $n$, define $f_n(x)$ to be the number of square-free positive integers $k$ such that $kx \le n$. Then, define $v_(n)$ as $$v(n) =\sum^n_{i=1}\sum^n_{j=1}f_n(i^2)- 6f_n (ij) + f_n(j^2).$$ Compute the largest positive integer $2 \le n \le 100$ for which $v(n)-v(n-1)$ is negative. (Note: A square-free positive integer is a positive integer that is not divisible by the square of any prime.)

Let $a_1,a_2,\ldots,a_n$ be a finite sequence of non negative integers, its subsequences are the sequences of the form $a_i,a_{i+1},\ldots,a_j$ with $1\le i\le j \le n$. Two subsequences are said to be equal if they have the same length and have the same terms, that is, two subsequences $a_i,a_{i+1},\ldots,a_j$ and $a_u,a_{u+1},\ldots a_v$ are equal iff $j-i=u-v$ and $a_{i+k}=a_{u+k}$ forall integers $k$ such that $0\le k\le j-1$. Finally, we say that a subsequence $a_i,a_{i+1},\ldots,a_j$ is palindromic if $a_{i+k}=a_{j-k}$ forall integers $k$ such that $0\le k \le j-i$ What is the greatest number of different palindromic subsequences that can a palindromic sequence of length $n$ contain?

Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8. Prove that the sum \[\sum_{i = 1}^8 P_{i - 1} P_i^2\] is minimal if and only if $Q$ is the centre of the circle.

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Circle $\omega$ inscribed in triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. On the extension of segment $AA_1$, point $A$ is taken as point D such that $AD= AC_1$. Lines $DB_1$ and $DC_1$ intersect a second time circle $\omega$ at points $B_2$ and $C_2$. Prove that $B_2C_2$ is the diameter of circle of $\omega$.

A rook stands in a corner of an $n$ by $n$ chess board. For what $n$, moving alternately along horizontals and verticals, can the rook visit all the cells of the board and return to the initial corner after $n^2$ moves? (A cell is visited only if the rook stops on it, those that the rook “flew over” during the move are not counted as visited.) Proposed by A. Spivak

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

In a round-robin table tennis tournament between $2010$ athletes, where each match ends with a winner and a loser, let $a_1,... , a_{2010}$ denote the number of wins of each athlete, and let $b_1, .., b_{2010}$ denote the number of losses of each athlete. Show that $a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}$.

There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

Determine whether there exists positive integers $a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}$ such that all sums $ a_{i}+a_{j}$, where 1 $\leq i < j \leq k$, are unique, and among those sums, there are $1000$ consecutive integers.

Estimate the range of the submissions for this problem. Your answer must be between $[0, 1000]$. An estimate $E$ earns $\frac{2}{1+0.05|A-E|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.2[/i]

Given that \begin{align*}x &= 1 - \frac 12 + \frac13 - \frac 14 + \cdots + \frac1{2007},\\ y &= \frac{1}{1005} + \frac{1}{1006} + \frac{1}{1007} + \cdots + \frac 1{2007},\end{align*} find the value of $k$ such that \[x = y + \frac 1k.\]

Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.

There are $N\ge3$ letters arranged in a circle, and each letter is one of $L$, $T$ and $F$. For a letter, we can do the following operation: if its neighbors are the same, then change it to the same letter too; otherwise, change it so that it is different from both its neighbors. Show that for any initial state, one can perform finitely many operations to achieve a stable state. Here, a stable state means that any operation does not change any of the $N$ letters. (ltf0501)

A grid is called $k$-special if in each cell is written a distinct integer such that the set of integers in the grid is precisely the set of positive divisors of $k$. A grid is called $k$-awesome if it is $k$-special and for each positive divisor $m$ of $k$, there exists an $m$-special grid within this $k$-special grid (within meaning you could draw a box in this grid to obtain the new grid). Find the sum of the $4$ smallest integers $k$ for which no $k$-awesome grid exists. [i]Proposed by Oliver Hayman[/i]

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

let $ x, y, z$ be real number in the interval $ [\frac12;2]$ and $ a, b, c$ a permutation of them. Prove the inequality: $ \dfrac{60a^2\minus{}1}{4xy\plus{}5z}\plus{}\dfrac{60b^2\minus{}1}{4yz\plus{}5x}\plus{}\dfrac{60c^2\minus{}1}{4zx\plus{}5y}\geq 12$

Let $ABC$ be an isosceles right triangle with hypotenuse $AB$, $D$ be some point in the plane such that $2CD = AB$ and point $C$ inside the triangle $ABD$. We construct two rays with a start in $C$, intersecting $AD$ and $BD$ and perpendicular to them. On the first one, intersecting $AD$, we will plot the segment $CK = AD$, and on the second one - $CM = BD$. Prove that points $M$, $D$ and $K$ lie on the same line.

Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$