Consider a regular heptagon ( polygon of $7$ equal sides and angles) $ABCDEFG$ as in the figure below:- $(a).$ Prove $\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}$ $(b).$ Using $(a)$ or otherwise, show that $\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}$ [asy] draw(dir(360/7)..dir(2*360/7),blue); draw(dir(2*360/7)..dir(3*360/7),blue); draw(dir(3*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(5*360/7),blue); draw(dir(5*360/7)..dir(6*360/7),blue); draw(dir(6*360/7)..dir(7*360/7),blue); draw(dir(7*360/7)..dir(360/7),blue); draw(dir(2*360/7)..dir(4*360/7),blue); draw(dir(4*360/7)..dir(1*360/7),blue); label("$A$",dir(4*360/7),W); label("$B$",dir(5*360/7),S); label("$C$",dir(6*360/7),S); label("$D$",dir(7*360/7),E); label("$E$",dir(1*360/7),E); label("$F$",dir(2*360/7),N); label("$G$",dir(3*360/7),W); [/asy]

A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to $ \textbf{(A)}\ 4.0\qquad \textbf{(B)}\ 4.2\qquad \textbf{(C)}\ 4.5\qquad \textbf{(D)}\ 5.0\qquad \textbf{(E)}\ 5.6$

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.

Let $n$ be a positive integer. Prove that there exists an integer $k$, $k\geq 2$, and numbers $a_i \in \{ -1, 1 \}$, such that \[ n = \sum_{1\leq i < j \leq k } a_ia_j . \]

Find $m$ and $n$ such that the set of points whose coordinates $x$ and $y$ satisfy the equation $|y-2x|=x$, coincides with the set of points specified by the equation $|mx + ny| = y$.

Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.

Points $ A \equal{} (3,9), B \equal{} (1,1), C \equal{} (5,3),$ and $ D \equal{} (a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ ABCD$. The quadrilateral formed by joining the midpoints of $ \overline{AB}, \overline{BC}, \overline{CD},$ and $ \overline{DA}$ is a square. What is the sum of the coordinates of point $ D$? $ \textbf{(A)} \ 7 \qquad \textbf{(B)} \ 9 \qquad \textbf{(C)} \ 10 \qquad \textbf{(D)} \ 12 \qquad \textbf{(E)} \ 16$

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases [b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$. [b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$. In other words, find the Steiner trees of the set $M$ in the above two cases.

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

The coordinate of $ P$ at time $ t$, moving on a plane, is expressed by $ x = f(t) = \cos 2t + t\sin 2t,\ y = g(t) = \sin 2t - t\cos 2t$. (1) Find the acceleration vector $ \overrightarrow{\alpha}$ of $ P$ at time $ t$ . (2) Let $ L$ denote the line passing through the point $ P$ for the time $ t%Error. "neqo" is a bad command. $, which is parallel to the acceleration vector $ \overrightarrow{\alpha}$ at the time. Prove that $ L$ always touches to the unit circle with center the origin, then find the point of tangency $ Q$. (3) Prove that $ f(t)$ decreases in the interval $ 0\leq t \leqq \frac {\pi}{2}$. (4) When $ t$ varies in the range $ \frac {\pi}{4}\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the figure formed by moving the line segment $ PQ$.

Let $ ABC$ be a right triangle with $ \angle A \equal{} 90^\circ$. Let $ D$ be the midpoint of $ AB$ and let $ E$ be a point on segment $ AC$ such that $ AD \equal{} AE$. Let $ BE$ meet $ CD$ at $ F$. If $ \angle BFC \equal{} 135^\circ$, determine $ BC / AB$.

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.

A cat and mouse live on a house mapped out by the points $(-1, 0)$, $(-1, 2)$, $(0, 3)$, $(1, 2)$, $(1, 0)$. The cat starts at the top of the house (point $(0, 3)$) and the mouse starts at the origin (0, 0). Both start running clockwise around the house at the same time. If the cat runs at $12$ units a minute and the mouse at 9 units a minute, how many laps around the house will the cat run before it catches the mouse?

A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer? [asy]size(12cm); pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); pair B = (-1,-1); pair C = (-1,-7); pair A = (-13,-7); path inclined_plane = A--B--C--cycle; draw(inclined_plane, p); real r = 1; // for marking angles draw(arc(A, r, 0, degrees(B-A))); // mark angle label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle real h = 1.2; // height of box real w = 1.9; // width of box path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box // box on slope with label picture box_on_slope; filldraw(box_on_slope, box, light_grey, black); label(box_on_slope, "$m$", (w/2,h/2)); pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB pair T1 = dir(125); // point of tangency with pulley pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box draw(T1--X1); // string add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope); // picture for the hanging box picture hanging_box; filldraw(hanging_box, box, light_grey, black); label(hanging_box, "$M$", (w/2,h/2)); pair T2 = (1,0); pair X2 = (1,-3); draw(T2--X2); // string add(shift(X2-(w/2,h)) * hanging_box); // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]

Triangle $ ABC$ has $ AC\equal{}3$, $ BC\equal{}4$, and $ AB\equal{}5$. Point $ D$ is on $ \overline{AB}$, and $ \overline{CD}$ bisects the right angle. The inscribed circles of $ \triangle ADC$ and $ \triangle BCD$ have radii $ r_a$ and $ r_b$, respectively. What is $ r_a/r_b$? $ \textbf{(A)}\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)$

The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.

Let $F$ be a smooth (i.e. $C^{\infty}$) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an [i]almost-immersion[/i] if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$, respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$. Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.

In the coordinate plane, define $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is defined on $M$, sends $(a,b)$ to $(a + b, b)$. Transformation $T$, also defined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we can use $S,T$ denitely to map it to $(g,0)$.

We are given two concurrent lines $ d_1 $ and $ d_2. $ Find, analytically, the acute angle formed by them such that for any point $ A $ the equation $ A=A_4 $ holds, where $ A_1 $ is the symmetric of $ A $ with respect to $ d_1, $ $ A_2 $ is the symmetric of $ A_1 $ with respect to $ d_2, $ $ A_3 $ is the symmetric of $ A_2 $ with respect to $ d_1, $ and $ A_4 $ is the symmetric of $ A_3 $ with respect to $ d_2. $

[b]a)[/b] Let be two nonzero complex numbers $ a,b. $ Show that the area of the triangle formed by the representations of the affixes $ 0,a,b $ in the complex plane is $ \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| . $ [b]b)[/b] Let be an equilateral triangle $ ABC, $ its circumcircle $ \mathcal{C} , $ its circumcenter $ O, $ and two distinct points $ P_1,P_2 $ in the interior of $ \mathcal{C} . $ Prove that we can form two triangles with sides $ P_1A,P_1B,P_1C, $ respectively, $ P_2A,P_2B,P_2C, $ whose areas are equal if and only if $ OP_1=OP_2. $

In $ n$-dimensional Euclidean space, the square of the two-dimensional Lebesgue measure of a bounded, closed, (two-dimensional) planar set is equal to the sum of the squares of the measures of the orthogonal projections of the given set on the $ n$-coordinate hyperplanes. [i]L. Tamassy[/i]

Let $u$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions: $\text{(i) }\frac{a}{2}\le x\le 2a\qquad\text{(ii) }\frac{a}{2}\le y\le 2a\qquad\text{(iii) }x+y\ge a \\ \\ \qquad\text{(iv) }x+a\ge y\qquad \text{(v) }y+a\ge x$ The boundary of set $S$ is a polygon with $\textbf{(A) }3\text{ sides}\qquad\textbf{(B) }4\text{ sides}\qquad\textbf{(C) }5\text{ sides}\qquad\textbf{(D) }6\text{ sides}\qquad \textbf{(E) }7\text{ sides}$

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?