Found problems: 85335
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers
Determine the number of subset $S$ of the set $T = {1, 2,..., 2005}$
such that the sum of elements in $s$ is congruent to 2006 modulo
2048.
Let $a_k$ be the number of ordered $10$-tuples $(x_1, x_2, ..., x_{10})$ of nonnegative integers such that
$$x^2_1+ x^2_2+ ... + x^2_{10} = k.$$
Let $b_k = 0$ if $a_k$ is even and $b_k = 1$ if $a_k$ is odd. Find $\sum^{2012}_{i=1} b_{4i}$.
For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many
If $a^2+b^2+c^2+d^2=1$, prove that $$(a-b)^2+(b-c)^2+(c-d)^2+(a-c)^2+(a-d)^2+(b-d)^2\leq 4$$
When does equality holds?
Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$. A point $P$ (other than $B$) is chosen on the segment $BC$ in such a way that $AB=AP$. Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$, and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$. Let $K$ be the intersection of lines $AP$ and $DE$. Let $F$ be a point on $BC$ (other than $P$) such that $KP=KF$. Show that $C,\ D,\ E$ and $F$ lie on the same circle.
The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than
(a) $0.7$ km?
(b) $0.8$ km?
(Grigory Kondakov, Moscow)
You may assume that the banks consist of segments and arcs of circles.
Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$
Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.
A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: $3$ points for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points.
(a) Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team?
(b) Is this possible if there are six teams in the tournament instead?
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
Let $P$ be an interior point of a triangle $ABC$ whose sidelengths are 26, 65, 78. The line through $P$ parallel to $BC$ meets $AB$ in $K$ and $AC$ in $L$. The line through $P$ parallel to $CA$ meets $BC$ in $M$ and $BA$ in $N$. The line through $P$ parallel to $AB$ meets $CA$ in $S$ and $CB$ in $T$. If $KL,MN,ST$ are of equal lengths, find this common length.
For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that
\[a_1 + 2a_2 + \cdots + ka_k = 1979.\]
Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.
Alice thinks about a natural number in her mind. Bob tries to find that number by asking him the following 10 questions:
[list]
[*]Is it divisible by 1?
[*]Is it divisible by 2?
[*]Is it divisible by 3?
[*]...
[*]Is it divisible by 9?
[*]Is it divisible by 10?
[/list]
Alice's answer to all questions except one was "yes". When she answers "no", she adds that "the greatest common factor of the number I have in mind and the divisor in the question you asked is 1”. According to this information, to which question did Alice answer "no"?
In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ Determine the exponent $\beta$, as large as possible such that for some $C$ independent of $n$
\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]
Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
Among citizens of Cambridge there exist $8$ different types of blood antigens. In a crowded lecture hall are $256$ students, each of whom has a blood type corresponding to a distinct subset of the antigens; the remaining of the antigens are foreign to them.
Quito the Mosquito flies around the lecture hall, picks a subset of the students uniformly at random,
and bites the chosen students in a random order. After biting a student, Quito stores a bit of any antigens that student had. A student bitten while Quito had $k$ blood antigen foreign to him/her will suffer for $k$ hours. What is the expected total suffering of all $256$ students, in hours?
Let $a_1, a_2, \ldots, a_{11}$ be integers. Prove that there exist numbers $b_1, b_2, \ldots, b_{11}$ such that
[list]
[*] $b_i$ is equal to $-1,0$ or $1$ for all $i \in \{1, 2,\dots, 11\}$.
[*] all numbers can't be zero at a time.
[*] the number $N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}$ is divisible by $2024$.
[/list]
In $\triangle{PQR}$, $PQ=4$, $PR=5$, and $QR=6$. Assume that an equilateral hexagon $ABCDEF$ is able to be drawn inside $\triangle{PQR}$ so that $\overline{AB}$ is parallel to $\overline{QR}$, $\overline{CD}$ is parallel to $\overline{PQ}$, $\overline{EF}$ is parallel to $\overline{RP}$, $\overline{BC}$ lies on $\overline{RP}$, $\overline{DE}$ lies on $\overline{QR}$, and $\overline{AF}$ lies on $\overline{PQ}$. Find the area of hexagon $ABCDEF$.
Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?
Let $a_1, a_2, ..., a_k$ be a finite arithmetic sequence with
\[ a_4+a_7+a_{10}=17 \] and \[ a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 \] If $a_k=13$, then $k=$
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
[asy]
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("$A$",A,W,linewidth(4));
dot("$B$",B,dir(0),linewidth(4));
dot("$C$",C,dir(0),linewidth(4));
dot("$D$",D,dir(20),linewidth(4));
dot("$E$",E,dir(100),linewidth(4));
dot("$F$",F,W,linewidth(4));
dot("$X$",X,dir(0),linewidth(4));
dot("$Y$",Y,N,linewidth(4));
dot("$Z$",Z,W,linewidth(4));
[/asy]
$ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$
$\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3}
\qquad\textbf{(E)}\ 21\sqrt{6}$
If $p,q,$ and $r$ are primes with $pqr=7(p+q+r)$, find $p+q+r$.