In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a \ni B, b \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).

Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.

Let $ABCD$ be a three-link broken line in space, all links of which are equal and $\angle BCD=90^o$. Find the distance from $A$ to the midpoint of $BD$, if $AD = a$.

Points $A, B, C$ and $D$ have these coordinates: $A(3,2), B(3,-2), C(-3,-2)$ and $D(-3, 0)$. The area of quadrilateral $ABCD$ is [asy] for (int i = -4; i <= 4; ++i) { for (int j = -4; j <= 4; ++j) { dot((i,j)); } } draw((0,-4)--(0,4),linewidth(1)); draw((-4,0)--(4,0),linewidth(1)); for (int i = -4; i <= 4; ++i) { draw((i,-1/3)--(i,1/3),linewidth(0.5)); draw((-1/3,i)--(1/3,i),linewidth(0.5)); }[/asy] $ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 18\qquad\text{(D)}\ 21\qquad\text{(E)}\ 24 $

For $i = 1, 2$ let $\vartriangle A_iB_iC_i$ be a triangle with side lengths $a_i, b_i, c_i$ and altitude lengths $p_i, q_i, r_i$. Define $a_3 =\sqrt{a_1^2 + a_2^2}, b_3 =\sqrt{b_1^2 + b_2^2}$ , and $c_3 =\sqrt{c_1^2 + c_2^2}$. Prove that $a_3, b_3, c_3$ are side lengths of a triangle, and if $p_3, q_3, r_3$ are the lengths of altitudes of this triangle, then $p_3^2 \ge p_1^2 +p_2^2$, $q_3^2 \ge q_1^2 +q_2^2$ , and $r_3^2 \ge r_1^2 +r_2^2$

Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$.

Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt {50}$ and $ BE \equal{} 1$. What is the area of the inner square $ EFGH$? [asy]unitsize(4cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair D=(0,0), C=(1,0), B=(1,1), A=(0,1); pair F=intersectionpoints(Circle(D,2/sqrt(5)),Circle(A,1))[0]; pair G=foot(A,D,F), H=foot(B,A,G), E=foot(C,B,H); draw(A--B--C--D--cycle); draw(D--F); draw(C--E); draw(B--H); draw(A--G); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,NNW); label("$F$",F,ENE); label("$G$",G,SSE); label("$H$",H,WSW);[/asy]$ \textbf{(A)}\ 25\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 42$

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

Circles $O_1$ and $O_2$ intersects at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. We join $AB$ to intersect $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume that $H$ is an arbitrary point on the line segment $AF$. We join $HE$ and extend it to intersect circle $O_1$ at point $G$, and join $BG$ and extend it to intersect the extended line of $AC$ at point $D$. Prove that $\frac{AH}{HF}=\frac{AC}{CD}$.

Let $P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)$ be points on the plane such that $P_0P_1P_2\Delta$ contains the origin $O$. Show that the areas of triangles $P_0OP_1,P_0OP_2,P_1OP_2$ form a geometric sequence in that order if and only if there exists a real number $x$, such that $$ a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0 $$

Point $O$ is interior to triangle $ABC$. Through $O$, draw three lines $DE \parallel BC, FG \parallel CA$, and $HI \parallel AB$, where $D, G$ are on $AB$, $I, F$ are on $BC$ and $E, H$ are on $CA$. Denote by $S_1$ the area of hexagon $DGHEFI$, and $S_2$ the area of triangle $ABC$. Prove that $S_1 \geq \frac 23 S_2.$

Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then (i) $ G$ is not simple; (ii) $ G$ contains a normal subgroup of order $ m$.

A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.

Consider the $ 12$-sided polygon $ ABCDEFGHIJKL$, as shown. Each of its sides has length $ 4$, and each two consecutive sides form a right angle. Suppose that $ \overline{AG}$ and $ \overline{CH}$ meet at $ M$. What is the area of quadrilateral $ ABCM$? [asy]unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,NE); label("$D$",D,NE); label("$E$",Ep,SE); label("$F$",F,SE); label("$G$",G,SE); label("$H$",H,SW); label("$I$",I,SW); label("$J$",J,SW); label("$K$",K,NW); label("$L$",L,NW); label("$M$",M,W);[/asy]$ \textbf{(A)}\ \frac {44}{3}\qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ \frac {88}{5}\qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ \frac {62}{3}$

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

Let $N_1$ be the answer to problem 1. On square $JHMT$, point $X$ lies on $\overline{HM}$, and point $Y$ is the intersection point of lines $JM$ and $TX$. Assume that $\tfrac{TY}{XY}=\sqrt5$ and the area of $\triangle{XYM}$ is $N_1$. Compute the area of $\triangle{JYT}$.

[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover: a) greater than $9$ ? b) greater than $15$? [b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] [b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color. [b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side? [b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

A wooden cube with edge length $ n$ units (where $ n$ is an integer $ >2$) is painted black all over. By slices parallel to its faces, the cube is cut into $ n^3$ smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $ n$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ \text{none of these}$

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent. [i]Pablo Jaén, Panama[/i]

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?