Found problems: 85335
$10$ teams participated in a football tournament: every two teams played each other exactly once. After the end of the tournament it turned out that all teams got different amount of points and some teams won more games, than the winner of the tournament, call them strong.
What is the maximum number of teams that could be strong? (In football the winner of the match gets three points, the loser - 0 points, and if the match ends in a draw both teams get 1 point.)
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.
A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.
Suppose that $4^{x_1}= 5, 5^{x_2}= 6,6^{x_3} = 7,..., 126^{x_{123}} = 127,127^{x_{124}} = 128$. What is the value of the product $X_1X_2... X_{124}$?
Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.
Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$.
Croatia
Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$. Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.
Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$
Evaluate the sum \[ \dfrac {1}{2 \lfloor \sqrt {1} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {2} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {3} \rfloor + 1} + \cdots + \dfrac {1}{2 \lfloor \sqrt {100} \rfloor + 1} \]
In the unit cube, given 75 points, no three of which are collinear. Prove that there exits a triangle whose vertices are among the given points and whose area is not greater than 7/72.
Find the number of $ x $ which follows the following :
$ x-\frac{1}{x}=[x]-[\frac{1}{x}] $
$ ( \frac{1}{100} \le x \le {100} ) $
Let $ABC$ be a triangle. $D$ and $E$ are points on line segments $BC$ and $AC$, respectively, such that $AD=60$, $BD=189$, $CD=36$, $AE=40$, and $CE=50$. What is $AB+DE$?
[i]2017 CCA Math Bonanza Tiebreaker Round #2[/i]
Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that
\[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\]
[i]Proposed by usjl[/i]
We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?
Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make?
[i]Proposed by Anthony Yang[/i]
Determine all real numbers $x$ which satisfy the inequality: \[ \sqrt{3-x}-\sqrt{x+1}>\dfrac{1}{2} \]
Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way:
(a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$.
(b) $B_{i+1}\neq C_i$ is the intersection of line $A_iC_i$ with the circumcircle of $O_iA_{i+1}C_i$.
(c) $C_{i+1}\neq B_i$ is the intersection of line $A_iB_i$ with the circumcircle of $O_iA_{i+1}B_i$.
Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \]
Note: $[K]$ denotes the area of $K$.
[i]Proposed by Yang Liu[/i]
In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$.
[asy]
unitsize(1 cm);
pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);
draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));
label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60));
[/asy]
Tomás is an avid domino player. One day, while playing with the tiles, he realized he could arrange all the tiles in a single row following the rules, meaning that the number on the right side of each tile matches the number on the left side of the next tile. If the number on the left side of the first tile is 5, what is the number on the right side of the last tile?
Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.
Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions:
a) $f(1)+1>0$;
b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$;
c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.
Let $n_1,n_2,\ldots,n_s$ be distinct integers such that
$$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample:
$(\text a)$ $|n_i|=1$ for some $i$
$(\text b)$ If further all $n_i$ are positive, then
$$\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.$$
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?