Found problems: 85335
Let $n \in N$ and $k$ be such that $1 \le k \le n$. Find the number of ordered $k$-tuples $(a_1, a_2,...,a_k)$ of integers such the $1 \le a_j \le n$, for $1 \le j \le k$ and [u]either [/u] there exist $l,m \in \{1, 2,..., k\}$ such that $l < m$ but $a_l > a_m$ [u]or [/u] there exists $l \in \{1, 2,..., k\}$ such that $a_l - l$ is an odd number.
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$.
[i]Proposed by Serbia[/i]
Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$.
[i]Proposed by James Lin.[/i]
In the $\triangle ABC$ shown, $D$ is some interior point, and $x, y, z, w$ are the measures of angles in degrees. Solve for $x$ in terms of $y, z$ and $w$.
[asy]
draw((0,0)--(10,0)--(2,7)--cycle);
draw((0,0)--(4,3)--(10,0));
label("A", (0,0), SW);
label("B", (10,0), SE);
label("C", (2,7), W);
label("D", (4,3), N);
label("x", (2.25,6));
label("y", (1.5,2), SW);
label("$z$", (7.88,1.5));
label("w", (4,2.85), S);
[/asy]
$ \textbf{(A)}\ w-y-z \qquad\textbf{(B)}\ w-2y-2z \qquad\textbf{(C)}\ 180-w-y-z \\
\qquad\textbf{(D)}\ 2w-y-z \qquad\textbf{(E)}\ 180-w+y+z $
Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that
$$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$
Show that:
$$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$
[i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]
In a plane we have $n$ lines, no two of which are parallel or perpendicular, and no three of which are concurrent. A cartesian system of coordinates is chosen for the plane with one of the lines as the $x$-axis. A point $P$ is located at the origin of the coordinate system and starts moving along the positive $x$-axis with constant velocity. Whenever $P$ reaches the intersection of two lines, it continues along the line it just reached in the direction that increases its $x$-coordinate. Show that it is possible to choose the system of coordinates in such a way that $P$ visits points from all $n$ lines.
For $a, b, c>0,$ and $k\geq1,$ prove that
\[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\]
Author: MIHALY BENCZE
Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?
Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true:
$(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.
Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.
[i]Carl Schildkraut, USA[/i]
A $ 6\times 6$ square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always [b]two[/b] congruent rectangles.
Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$.
(1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin.
(2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized.
50 points
If the graph is a graph of POSITION vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) From A to B
(B) From B to C only
(C) From B to D
(D) From C to D only
(E) From D to E
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$
\[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \]
Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.
Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression.
[hide] be carefull ! :lol: [/hide]
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]
Let $\alpha \in \left( 0, \dfrac{\pi}{2}\right)$.Find the minimum value of the expression
$$ P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .$$
Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$
$$f(xf(y) - yf(x)) = f(xy) - xy.$$
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i]
For what natural number $n> 100$ can $n$ pairwise distinct numbers be arranged on a circle such that each number is either greater than $100$ numbers following it clockwise or less than all of them? and would any property be violated when deleting any of those numbers?
A regular $2004$-sided polygon is given, with all of its diagonals drawn. After some sides and diagonals are removed, every vertex has at most five segments coming out of it. Prove that one can color the vertices with two colors such that at least $\frac{3}{5}$ of the remaining segments have ends with different colors.
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]