Found problems: 85335
On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?
Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.
Solve the equation for positive integers $m, n$:
\[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]
Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an $m\times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\dbinom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$.
[i]Proposed by James Lin
Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.
Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.
Let $m$ be the maximum possible value of $x^{16} + \frac{1}{x^{16}}$, where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when $m$ is divided by $2007$.
Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that
\[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
A sphere with unit radius is given. Furthermore, circles $k_0,k_1,\ldots,k_n\ (n\ge3)$ of the same radius $r$ are given on the sphere. The circle $k_0$ is tangent to all other circles $k_i$ and every two circles $k_i,k_{i+1}$ are tangent for $i=1,\ldots,n$ (assuming $k_{n+1}=k_1$).
a) Find relation between numbers $n,r.$
b) Determine for which $n$ the described situation can occur and compute the corresponding radius $r.$
(We say non-planar circles are tangent if they have only a single common point and their tangent lines in this point coincide.)
Initially, the number $1$ is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations:
Operation A: Multiply the number on the board with $\frac12$.
Operation B: Subtract the number on the board from $1$.
For example, if the number $\frac38$ is on the board, it can be replaced by $\frac12 \frac38=\frac{3}{16}$ or by $1-\frac38=\frac58$ .
Give a sequence of steps after which the number on the board is $\frac{2009}{2^{20009}}$ .
The diagonals $AD$, $BE$, $CF$ of cyclic hexagon $ABCDEF$ intersect in $S$ and $AB$ is parallel to $CF$ and lines $DE$ and $CF$ intersect each other in $M$. Let $N$ be a point such that $M$ is the midpoint of $SN$. Prove that circumcircle of $\triangle ADN$ is passing through midpoint of segment $CF$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.
Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$.
If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.
What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$?
$ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$
Chim Tu has a large rectangular table. On it, there are finitely many pieces of paper with nonoverlapping interiors, each one in the shape of a convex polygon. At each step, Chim Tu is allowed to slide one piece of paper in a straight line such that its interior does not touch any other piece of paper during the slide. Can Chim Tu always slide all the pieces of paper off the table in finitely many steps?
Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.
Let $C$ be the curve $y^2 = x^3$ (where $x$ takes all non-negative real values). Let $O$ be the origin, and $A$ be the point where the gradient is $1.$ Find the length of the curve from $O$ to $A.$
In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively. Then the measure of $AB$ is equal to
$\textbf{(A) }\dfrac{1}{2}a+2b\qquad\textbf{(B) }\dfrac{3}{2}b+\dfrac{3}{4}a\qquad\textbf{(C) }2a-b\qquad\textbf{(D) }4b-\dfrac{1}{2}a\qquad \textbf{(E) }a+b$
[asy]
size(175);
defaultpen(linewidth(0.8));
real r=50, a=4,b=2.5,c=6.25;
pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D;
draw(A--B--C--D--cycle);
label("$A$",A,SW);
label("$B$",B,N);
label("$C$",C,E);
label("$D$",D,S);
label("$a$",D/2,N);
label("$b$",(C+D)/2,NW);
//Credit to djmathman for the diagram[/asy]
A [i]permutation[/i] of a finite set is a one-to-one function from the set onto itself. A [i]cycle[/i] in a permutation $P$ is a nonempty sequence of distinct items $x_1$, $\ldots\,$, $x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots\,$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1) = x_2$ and $P(x_2) = x_1$. Every permutation of a finite set splits the set into a finite number of disjoint cycles. If this number equals 2, then the permutation is called [i]bi-cyclic[/i]. Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".
Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Then one interior angle of the triangle, in degrees, is:
$\textbf{(A) }57\tfrac12\qquad
\textbf{(B) }59\qquad
\textbf{(C) }60\qquad
\textbf{(D) }61\qquad
\textbf{(E) }122$
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$.
Proposed by Karl Czakler
Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and
\[ g(x) h(x)
= \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \]
holds for all nonzero real numbers $x$. Find $g(2)$.
[i]Proposed by Yang Liu[/i]
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .