Found problems: 1581
Let the coordinate planes have the reflection property. A ray falls onto one of them. How does the final direction of the ray after reflecting from all three coordinate planes depend on its initial direction?
Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$, and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$. Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.
Given points $A,B,C$ and $D$ lie a circle. $AC\cap BD=K$. $I_1, I_2,I_3$ and $I_4$ incenters of $ABK,BCK,CDK,DKA$. $M_1,M_2,M_3,M_4$ midpoints of arcs $AB,BC,CA,DA$ . Then prove that $M_1I_1,M_2I_2,M_3I_3,M_4I_4$ are concurrent.
Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
In the non-isosceles triangle $ABC$ an altitude from $A$ meets side $BC$ in $D$ . Let $M$ be the midpoint of $BC$ and let $N$ be the reflection of $M$ in $D$ . The circumcirle of triangle $AMN$ intersects the side $AB$ in $P\ne A$ and the side $AC$ in $Q\ne A$ . Prove that $AN,BQ$ and $CP$ are concurrent.
Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$
a) Prove that $EF=\frac{BC}{2}.$
b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$.
Prove that lines $ AA_1,BB_1,CC_1$ are concurrent.
$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real theta = -100, r = 0.3; pair D2 = (0.3,0.76);
string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare);
for(int i = 0; i < lbl.length; ++i) {
pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5;
label("$"+lbl[i]+"'$", P, Q);
label("$"+lbl[i]+"$",D2+rotate(theta)*(r*P), rotate(theta)*Q);
}[/asy]
Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$.
[i]Author: L. Emelyanov[/i]
On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?
Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find:
a) The minimum value of $n$.
b) For that value, find the least possible number of red squares.
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle.
(a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$?
(b) What is the maximum area of this triangle?
Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.
(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points $ A$ and $ B$. Points $ X$, $ Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $ AX$ and $ BY$.
Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.
$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then
\[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]
Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.
Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$ . Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$.
a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line.
b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
Let $AA',BB', CC'$ be the bisectors of the angles of a triangle $ABC \ (A' \in BC, B' \in CA, C' \in AB)$. Prove that each of the lines $A'B', B'C', C'A'$ intersects the incircle in two points.