Found problems: 1581
Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$.
[i]Proposed by Dusan Djukic[/i]
Triangle $ABC$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point of reflection of $C$ about $AB$. It is known that $P$ and two midpoints of two sides of $ABC$ lie on a line. Find the angles of the triangle.
Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.
A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest.
[asy]
size(225);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.7));
draw((0,-1)--(2,-1),EndArrow);
label("$\vec{F}$",(1, -1),S);
label("Disk",(-1,0),W);
filldraw(circle((5,0),1),gray(.7));
filldraw(circle((5,0),0.75),white);
draw((5,-1)--(7,-1),EndArrow);
label("$\vec{F}$",(6, -1),S);
label("Hoop",(6,0),E);
filldraw(circle((10,0),1),gray(.5));
draw((10,-1)--(12,-1),EndArrow);
label("$\vec{F}$",(11, -1),S);
label("Sphere",(11,0),E);
[/asy]
$ \textbf{(A)} \ \text{disk, hoop, sphere}$
$\textbf{(B)}\ \text{sphere, disk, hoop}$
$\textbf{(C)}\ \text{hoop, sphere, disk}$
$\textbf{(D)}\ \text{disk, sphere, hoop}$
$\textbf{(E)}\ \text{hoop, disk, sphere} $
Let the line
$ z\equal{}x, \, y\equal{}0$
rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed.
(a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped).
(b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.
On each square of an $n\times n$-chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 10)[/i]
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
For each positive integer $ n$, let $ c(n)$ be the largest real number such that
\[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\]
for all triples $ (f, a, b)$ such that
--$ f$ is a polynomial of degree $ n$ taking integers to integers, and
--$ a, b$ are integers with $ f(a) \neq f(b)$.
Find $ c(n)$.
[i]Shaunak Kishore.[/i]
Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.
The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$?
$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$
Find all pairs of real numbers $a, b$ for which the equation in the domain of the real numbers
\[\frac{ax^2-24x+b}{x^2-1}=x\]
has two solutions and the sum of them equals $12$.
Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.
Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?
Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0\equal{}AD\cap BC$, $ B_0\equal{}BD\cap AC$, $ C_0 \equal{}CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}$.
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.
a) Prove that $B_{n}$ does not depend on location of $P$.
b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions:
$i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$.
$ii)$ There are no two lines of $S$ which are parallel.
Squares $ABB_{1}A_{2}$ and $BCC_{1}B_{2}$ are externally drawn on the hypotenuse $AB$ and on the leg $BC$ of a right triangle $ABC$ . Show that the lines $CA_{2}$ and $AB_{2}$ meet on the perimeter of a square with the vertices on the perimeter of triangle $ABC .$
A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?
Trapezoid $ ABCD$, with $ \overline{AB}\parallel{}\overline{CD}$, is inscribed in circle $ \omega$ and point $ G$ lies inside triangle $ BCD$. Rays $ AG$ and $ BG$ meet $ \omega$ again at points $ P$ and $ Q$, respectively. Let the line through $ G$ parallel to $ \overline{AB}$ intersects $ \overline{BD}$ and $ \overline{BC}$ at points $ R$ and $ S$, respectively. Prove that quadrilateral $ PQRS$ is cyclic if and only if $ \overline{BG}$ bisects $ \angle CBD$.
Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.