Found problems: 85335
At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Prove that there are two couples $gb$ and $g'b'$ which dance whereas $b$ does not dance with $g'$ nor does $g$ dance with $b'$.
Solve the equation in nonnegative integers $a,b,c$:
$3^a+2^b+2015=3c!$
I.Gorodnin
Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence.
[i]Moldavia Olympiad, 1994[/i]
Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.
Consider the triangle $ABC$ and the points $D \in (BC)$ and $M \in (AD)$. Lines $BM$ and $AC$ meet at $E$, lines $CM$ and $AB$ meet at $F$, and lines $EF$ and $AD$ meet at $N$. Prove that $$\frac{AN}{DN}=\frac{1}{2}\cdot \frac{AM}{DM}$$
Let $A$, $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$. Find the locus of $C'$.
Let $P$ be any point inside an acute triangle. Let $D$ and $d$ be respectively the maximum and minimum distances from $P$ to any point on the perimeter of the triangle.
(a) Prove that $D \ge 2d$.
(b) Determine when equality holds
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold:
[list=disc]
[*]The sides of each rectangle are parallel to the sides of the unit square.
[*]Each point in $S$ is [i]not[/i] in the interior of any rectangle.
[*]Each point in the interior of the unit square but [i]not[/i] in $S$ is in the interior of at least one of the $k$ rectangles
[/list]
(The interior of a polygon does not contain its boundary.)
[i]Holden Mui[/i]
Let $(a_n)_{n\ge1}$ be a sequence given by
\begin{align*}
a_1 &= 1, \\
a_{2^k+j} &= -a_j\text{ for any } k\ge0,1\le j\le 2^k.
\end{align*}
Show that the sequence is not periodic.
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$
[b]p1.[/b] Show that for every $n \ge 6$, a square in the plane may be divided into $n$ smaller squares, not necessarily all of the same size.
[b]p2.[/b] Let $n$ be the $4018$-digit number $111... 11222...2225$, where there are $2008$ ones and $2009$ twos. Prove that $n$ is a perfect square. (Giving the square root of $n$ is not sufficient. You must also prove that its square is $n$.)
[b]p3.[/b] Let $n$ be a positive integer. A game is played as follows. The game begins with $n$ stones on the table. The two players, denoted Player I and Player II (Player I goes first), alternate in removing from the table a nonzero square number of stones. (For example, if $n = 26$ then in the first turn Player I can remove $1$ or $4$ or $9$ or $16$ or $25$ stones.) The player who takes the last stone wins. Determine if the following sentence is TRUE or FALSE and prove your answer:
There are infinitely many starting values n such that Player II has a winning strategy.
(Saying that Player II has a winning strategy means that no matter how Player I plays, Player II can respond with moves that lead to a win for Player II.)
[b]p4.[/b] Consider a convex quadrilateral $ABCD$. Divide side $AB$ into $8$ equal segments $AP_1$, $P_1P_2$, $...$ , $P_7B$. Divide side $DC$ into $8$ equal segments $DQ_1$, $Q_1Q_2$, $...$ , $Q_7C$. Similarly, divide each of sides $AD$ and $BC$ into $8$ equal segments. Draw lines to form an $8 \times 8$ “checkerboard” as shown in the picture. Color the squares alternately black and white.
(a) Show that each of the $7$ interior lines $P_iQ_i$ is divided into $8$ equal segments.
(b) Show that the total area of the black regions equals the total area of the white regions.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/027f02e26613555181ed93d1085b0e2de43fb6.png[/img]
[b]p5.[/b] Prove that exactly one of the following two statements is true:
A. There is a power of $10$ that has exactly $2008$ digits in base $2$.
B. There is a power of $10$ that has exactly $2008$ digits in base $5$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Define $a_0 = 2$ and $a_{n+1} = a^2_n + a_n -1$ for $n \ge 0$. Prove that $a_n$ is coprime to $2n + 1$ for all $n \in N$.
Evaluate
\[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\]
Own
The area of the largest square that can be inscribed in a regular hexagon with sidelength $1$ can be expressed as $a-b\sqrt{c}$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.
A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$. Find the area of the square formed by drawing the chords $AF$ , $BE$, $CH$ and $DG$.
$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?
$ \textbf{(A)} \ 55 \qquad \textbf{(B)} \ 56 \qquad \textbf{(C)} \ 57 \qquad \textbf{(D)} \ 58 \qquad \textbf{(E)} \ 59$
Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that
\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]
Find all polynomials $P(x)$ with real coefficients that satisfy the identity $$P(x^3-2)=P(x)^3-2,$$ for every real number $x$.
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$.
( V Senderov)
10) The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the bottom of the jug? $\rho$ is the density of the milk.
A) $P = \rho gh$
B) $P = \rho gH$
C) $\rho gH< P < \rho gh$
D) $P > \rho gh$
E) $P < \rho gH$
Find the number of ordered pairs of positive integer solutions $(m,n)$ to the equation $20m+12n=2012.$
On a circle, $n$ points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?
Positive integers $a,b,c$ satisfying the equation
$$a^3+4b+c = abc,$$
where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.
Let $a,b,c $be positive real numbers.Prove that
$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.