[b]6.1[/b] Which number is greater $$\underbrace{1000. . . 001}_{1965\, zeroes} / \underbrace{1000 . . . 001}_{1966\, zeroes} \,\,\, or \,\,\, \underbrace{1000. . . 001}_{1966\, zeroes} / \underbrace{1000 . . . 001}_{1967\, zeroes} \,\,?$$ [b]6.2[/b] $30$ teams participate in the football championship. Prove that at any moment there will be two teams that have played at this point the same number of matches. [b]6.3./ 7.1 [/b] All integers from $1$ to $1966$ are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]6.4 / 7.5[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]6.5[/b] In a chess tournament, there are more than three chess players, and each player plays each other the same number of times. There were $26$ rounds in the tournament. After the $13$th round, one of the participants discovered that he had an odd number points, and each of the other participants has an even number of points. How many chess players participated in the tournament? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

It is known that in a group of $2024$ students each student has at least $1011$ acquaintances among the remaining members of the group. What is more, there exists a student that has at least $1012$ acquaintances in the group. Prove that for every pair of students $X, Y$, there exist students $X_0 = X, X_1, ..., X_{n - 1}, X_n = Y$ in the group such that for every index $i = 0, ..., n - 1$, the students $X_i$ and $X_{i + 1}$ are acquaintances. [i]Proposed by Mirko Petruševski[/i]

a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that $\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$ for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$. b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that $\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$ for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.

Compute the number of permutations of $\{1,2,3\}$ with the property that there is some number that can be removed such that the remaining numbers are in increasing order. For example, $(2,1,3)$ has this property because removing $1$ leaves $(2,3)$, which is in increasing order. [i]2020 CCA Math Bonanza Team Round #1[/i]

Let the circles $\omega$ and $\omega'$ intersect in points $A$ and $B$. The tangent to circle $\omega$ at $A$ intersects $\omega'$ at $C$ and the tangent to circle $\omega'$ at $A$ intersects $\omega$ at $D$. Suppose that the internal bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $E$ and $F$, respectively, and the external bisector of $\angle CAD$ intersects $\omega$ and $\omega'$ at $X$ and $Y$, respectively. Prove that the perpendicular bisector of $XY$ is tangent to the circumcircle of triangle $BEF$. [i]Proposed by Mahdi Etesami Fard[/i]

The set $S$ contains nine numbers. The mean of the numbers in $S$ is $202.$ The mean of the five smallest of the numbers in $S$ is $100.$ The mean of the five largest numbers in $S$ is $300.$ What is the median of the numbers in $S?$

There are $ n$ football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let $ k$ be as follows: $ 2 \leq k \leq n \minus{} 1$. At least how many points must a certain team get in the competition so as to ensure that there are at most $ k \minus{} 1$ teams whose scores are not less than that particular team's score?

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel? $ \textbf{(A)}\ 425 \qquad \textbf{(B)}\ 444 \qquad \textbf{(C)}\ 456 \qquad \textbf{(D)}\ 484 \qquad \textbf{(E)}\ 506$

Let $G$ be a finite set of real $n \times n$ matrices $\left\{M_{i}\right\}, 1 \leq i \leq r,$ which form a group under matrix multiplication. Suppose that $\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0,$ where $\operatorname{tr}(A)$ denotes the trace of the matrix $A .$ Prove that $\textstyle\sum_{i=1}^{r} M_{i}$ is the $n \times n$ zero matrix.

Let $R$ be the rectangle on the cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$. [i]Proposed by Boyan Litchev[/i] [hide=Solution][i]Solution[/i]. $\boxed{85}$ We have $(6-n)(8-n)$ distinct squares with side length $n$, so the total number of squares is $5 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}$.[/hide]

In an urn there are one ball marked $1$, two balls marked $2$, and so on, up to $n$ balls marked $n$. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.

Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for$ 8$ games. Who rested in the last game?

Given a $ n \times n$ grid board . How many ways can an $X$ and an $O$ be placed in such a way that they are not in adjacent squares?

Let $ABC$ be an acute triangle with $\angle BAC=30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$, respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$, respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$. Prove that $\angle BPC=90^{\circ}$ .

The number \[ \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} \] can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $a\cdot b\cdot c$.

Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.

Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let the areas of triangles $ABC$, $ACD$, and $ADB$ be denoted by $x$, $y$, and $z$, respectively. In terms of $x$, $y$, and $z$, find the area of triangle $BCD$.

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point $(1,1,1)$ and (b) if the tangent plane is drawn at any point $P$ and $X,Y, Z$ are the intersections of this plane with the $x, y$ and $z-$axis respectively, then $P$ is the orthocenter of the triangle $XYZ.$

If $m,n,p,q\in\mathbb N,m,n,p,q\ge4$ then prove that: $$4^n(4^n+1)+4^m(4^m+1)+4^p(4^p+1)+4^q(4^q+1)\ge4mnpq(mnpq+1)$$ [i]Proposed by Daniel Sitaru[/i]

In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$,$\overline{BC}$,$\overline{CD}$,$\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence? [asy] size(150); defaultpen(linewidth(0.8)); string[] strng = {'A','D','B','E','C'}; pair A=dir(90),B=dir(306),C=dir(162),D=dir(18),E=dir(234); draw(A--B--C--D--E--cycle); for(int i=0;i<=4;i=i+1) { path circ=circle(dir(90-72*i),0.125); unfill(circ); draw(circ); label("$"+strng[i]+"$",dir(90-72*i)); } [/asy] $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 11\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 13$

Let $ a_1<a_2<\cdots<a_n $ be positive integers such that for every distinct $1\leq{i,j}\leq{n}$ we have $ a_j-a_i $ divides $ a_i $. Prove that \[ ia_j\leq{ja_i} \qquad \text{ for } 1\leq{i}<j\leq{n} \]

Solve the equation $$\sqrt{2+\sqrt{2-\sqrt{2+x}}}=x.$$

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

If a polynomial of degree n has integer values when evaluated in each of $k,k+1,\ldots,k+n$, where $k$ is an integer, prove that the polynomial has integer values when evaluated at each integer $x$.