Found problems: 1342
Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that:
a) $ABCD$ is circumscriptible.
b) $MNPQ$ is cyclic.
c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.
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.
Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that
$ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.
Let $ ABC$ be an acute angled triangle; let $ D,F$ be the midpoints of $ BC,AB$ respectively. Let the perpendicular from $ F$ to $ AC$ and the perpendicular from $ B$ ti $ BC$ meet in $ N$: Prove that $ ND$ is the circumradius of $ ABC$.
[15 points out of 100 for the 6 problems]
On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.
[4] Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle{BDF}$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.
The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?
A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is $3$ more than the number of small rectangles.
In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.
[asy]unitsize(2.5mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
dotfactor=3;
pair O1=(0,0), O2=(12,0);
path C1=Circle(O1,8), C2=Circle(O2,6);
pair P=intersectionpoints(C1,C2)[0];
path C3=Circle(P,sqrt(130));
pair Q=intersectionpoints(C3,C1)[0];
pair R=intersectionpoints(C3,C2)[1];
draw(C1);
draw(C2);
//draw(O2--O1);
//dot(O1);
//dot(O2);
draw(Q--R);
label("$Q$",Q,N);
label("$P$",P,dir(80));
label("$R$",R,E);
//label("12",waypoint(O1--O2,0.4),S);[/asy]
Given rectangle $ R_1$ with one side $ 2$ inches and area $ 12$ square inches. Rectangle $ R_2$ with diagonal $ 15$ inches is similar to $ R_1.$ Expressed in square inches the area of $ R_2$ is:
$ \textbf{(A)}\ \frac92 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ \frac{135}{2} \qquad
\textbf{(D)}\ 9 \sqrt{10} \qquad
\textbf{(E)}\ \frac{27 \sqrt{10}}{4}$
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$- and $y$- axes, and call a set of points $nice$ if no two of them have the same $x$- or $y$- coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$.
[i]Yannick Yao[/i]
Circles $A$, $B$, and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circle $A$ and circle $B$?
[asy]
size(170); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3;
filldraw(arc((1,0),1,90,180)--arc((-1,0),1,0,90)--arc((0,1), 1, 180, 0)--cycle,gray);
draw(circle((0,1),1)); draw(circle((1,0),1)); draw(circle((-1,0),1));
dot((-1,0)); dot((1,0)); dot((0,1));
label("$A$",(-1,0),SW); label("$B$",(1,0),SE); label("$C$",(0,1),N);[/asy]
$ \textbf{(A)}\ 3-\frac{\pi}{2} \qquad
\textbf{(B)}\ \frac{\pi}{2} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{3\pi}{4} \qquad
\textbf{(E)}\ 1+\frac{\pi}{2}$
In rectangle $ ABCD$, $ AD \equal{} 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
dotfactor=4;
pair D=(0,0), C=(sqrt(3),0), B=(sqrt(3),1), A=(0,1), P=(sqrt(3)/3,1);
pair[] dotted={A,B,C,D,P};
draw(A--B--C--D--cycle);
draw(B--D--P);
dot(dotted);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$P$",P,N);[/asy]$ \textbf{(A)}\ 3 \plus{} \frac {\sqrt3}{3} \qquad\textbf{(B)}\ 2 \plus{} \frac {4\sqrt3}{3}\qquad\textbf{(C)}\ 2 \plus{} 2\sqrt2\qquad\textbf{(D)}\ \frac {3 \plus{} 3\sqrt5}{2} \qquad\textbf{(E)}\ 2 \plus{} \frac {5\sqrt3}{3}$
Let $ABC$ be a triangle with circumcenter $O$ such that $AC = 7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$.
Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.
Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior?
$ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $
Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then \[\angle DBC+\angle DPC = \angle DQC.\]
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that:
[list]
[*] every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color;
[*] every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color;
[*] every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color;
[*] every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color;
[*] no two neighboring vertices have been colored in $1.$ and $3.$ color;
[*] no two neighboring vertices have been colored in $2.$ and $4.$ color.
[/list]
Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).
In a $100\times25$ rectangle table, fill in a positive real number in each blank. Let the number in the $i$th line, the $j$th column be $x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)$ (shown in Fig.1 ). Then, we rearrange the numbers in each column: $x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)$ (shown in Fig.2 ). Find the minumum value of $k$, satisfying:
As long as $\sum_{j=1}^{25}x_{i,j}\leq1$ for numbers in Fig.1 ($i=1,2,\cdots,100$), then $\sum_{j=1}^{25}x'_{i,j}\leq1$ for $i\geq k$ in Fig.2.
$$\textbf{Fig.1}\\
\begin{tabular}{|c|c|c|c|}
\hline
$x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\
\hline
$x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\
\hline
$\cdots$&$\cdots$&$\cdots$&$\cdots$\\
\hline
$x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\
\hline
\end{tabular}
\qquad\textbf{Fig.2}\\
\begin{tabular}{|c|c|c|c|}
\hline
$x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\
\hline
$x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\
\hline
$\cdots$&$\cdots$&$\cdots$&$\cdots$\\
\hline
$x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\
\hline
\end{tabular}$$
Consider an 8x8 board divided in 64 unit squares. We call [i]diagonal[/i] in this board a set of 8 squares with the property that on each of the rows and the columns of the board there is exactly one square of the [i]diagonal[/i]. Some of the squares of this board are coloured such that in every [i]diagonal[/i] there are exactly two coloured squares. Prove that there exist two rows or two columns whose squares are all coloured.