Found problems: 85335
We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw?
Prove that for every positive integer $n$, the decimal expression of $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is periodic .
A positive integer $n$ is called[i] bad [/i]if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100. $
[i]Proposed by Michael Ren[/i]
Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.
Find the number of $10$-digit numbers $\overline{a_1a_2... a_{10}}$ which are multiples of $11$ such that the digits are non-increasing from left to right, i.e. $a_i \ge a_{i+1}$ for each $1 \le i \le 9$.
The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$.
Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$.
[i](A. Voidelevich)[/i]
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.
A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$. Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$
Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$
Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying
\[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\]
Show that for any real number $\alpha$, we have
\[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\]
Here $[\beta]$ denotes the greastest integer not larger than $\beta$.
Consider an equilateral triangle with side length $9$. Each side is divided into $3$ equal segments by $2$ points, for a total of $6$ points. Compute the area of the circle passing through these$ 6$ points.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/1860a3ff86a0e4b93a4891861300dcb09adad4.png[/img]
Let $ \mathcal{H} \equal{} A_1A_2\ldots A_n$ be a convex $ n$-gon. For $ i \equal{} 1, 2, \ldots, n$, let $ A'_{i}$ be the point symmetric to $ A_i$ with respect to the midpoint of $ A_{i \minus{} 1}A_{i \plus{} 1}$ (where $ A_{n \plus{} 1} \equal{} A_1$). We say that the vertex $ A_i$ is [i]good[/i] if $ A'_{i}$ lies inside $ \mathcal{H}$. Show that at least $ n \minus{} 3$ vertices of $ \mathcal{H}$ are [i]good[/i].
In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.
\[\begin{array}{c@{\hspace{8em}}
c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{4pt}}c@{\hspace{2pt}}
c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{6pt}}
c@{\hspace{6pt}}c@{\hspace{6pt}}c} \vspace{4pt}
\text{Row 0: } & & & & & & & 1 & & & & & & \\\vspace{4pt}
\text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\vspace{4pt}
\text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\vspace{4pt}
\text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt}
\text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt}
\text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\vspace{4pt}
\text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1
\end{array}\]
In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?
Find all polynomials $P(x)$ with real coefficients such that
\[(x-2010)P(x+67)=xP(x) \]
for every integer $x$.
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.
[i]Author: Christopher Bradley, United Kingdom [/i]
Solve the system of equations $\begin{cases} xy=1 \\
\frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\end{cases}$
Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.
In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.
Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$.
Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds
$f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$
[i]Proposed by Matija Bucić[/i]
A man has three daughters. The product of their ages is $168$, and he remembers that the sum of their ages is the number of trees in his yard. He counts the trees but cannot determine any of their ages. What are all possible ages of his oldest daughter?
2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .
$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$
Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that:
$\angle ANC + \angle ABC = 180$