The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half. (Anton Trygub)

We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.

Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$, set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$. Prove that \[S \leq \frac{1}{2}(x_1x_2+x_3x_4).\]

[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions. [b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale. [b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number? [b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it? [b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get? [img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .

Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively. The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$. Shows that $KM$ cannot be paralel to $IH$. PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect. PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students. Sorry if I had some English mistakes here.

Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that: $\bullet$ The area of square $KLMN$ is $1$. $\bullet$ The vertices of the triangle divide three sides of the square up into these ratios: $KX : XL = 3 : 2$ $KY : YN = 4 : 1$ $NZ : ZM = 2 : 3$ What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale). [img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]

Circle $\omega$ inscribed in triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$ and $C_1$ respectively. On the extension of segment $AA_1$, point $A$ is taken as point D such that $AD= AC_1$. Lines $DB_1$ and $DC_1$ intersect a second time circle $\omega$ at points $B_2$ and $C_2$. Prove that $B_2C_2$ is the diameter of circle of $\omega$.

Let $ \triangle ABC$ be a triangle with incircle $ \omega(I,r)$and circumcircle $ \zeta(O,R)$.Let $ l_{a}$ be the angle bisector of $ \angle BAC$.Denote $ P\equal{}l_{a}\cap\zeta$.Let $ D$ be the point of tangency $ \omega$ with $ [BC]$.Denote $ Q\equal{}PD\cap\zeta$.Show that $ PI\equal{}QI$ if $ PD\equal{}r$.

Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$

The circles $S_{1}$ and $S_{2}$ intersect at $M$ and $N$.Show that if vertices $A$ and $C$ of a rectangle $ABCD$ lie on $S_{1}$ while vertices $B$ and $D$ lie on $S_{2}$,then the intersection of the diagonals of the rectangle lies on the line $MN$.

(a) Is it possible to place five wooden cubes in space so that each of them has a part of its face touching each of the others? (b) Answer the same question, but with $6$ cubes.

Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$? $\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.

The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]D1.[/b] Triangle $ABC$ has $AB = 3$, $BC = 4$, and $\angle B = 90^o$. Find the area of triangle $ABC$. [b]D2 / L1.[/b] Let $ABCDEF$ be a regular hexagon. Given that $AD = 5$, find $AB$. [b]D3.[/b] Caroline glues two pentagonal pyramids to the top and bottom of a pentagonal prism so that the pentagonal faces coincide. How many edges does Caroline’s figure have? [b]D4 / L3.[/b] The hour hand of a clock is $6$ inches long, and the minute hand is $10$ inches long. Find the area of the region swept out by the hands from $8:45$ AM to $9:15$ AM of a single day, in square inches. [b]D5 / L2.[/b] Circles $A$, $B$, and $C$ are all externally tangent, with radii $1$, $10$, and $100$, respectively. What is the radius of the smallest circle entirely containing all three circles? [b]D6.[/b] Four parallel lines are drawn such that they are equally spaced and pass through the four vertices of a unit square. Find the distance between any two consecutive lines. [b]D7 / L4.[/b] In rectangle $ABCD$, $AB = 2$ and $AD > AB$. Two quarter circles are drawn inside of $ABCD$ with centers at $A$ and $C$ that pass through $B$ and $D$, respectively. If these two quarter circles are tangent, find the area inside of $ABCD$ that is outside both of the quarter circles. [b]D8 / L6.[/b] Triangle $ABC$ is equilateral. A circle passes through $A$ and is tangent to side $BC$. It intersects sides AB and $AC$ again at $E$ and $F$, respectively. If $AE = 10$ and $AF = 11$, find $AB$. [b]L5.[/b] Find the area of a triangle with side lengths $\sqrt{2}$, $\sqrt{58}$, and $2\sqrt{17}$. [b]L7.[/b] Triangle $ABC$ has area $80$. Point $D$ is in the interior of $\vartriangle ABC$ such that $AD =6$, $BD = 4$, $CD = 16$, and the area of $\vartriangle ADC = 48$. Determine the area of $\vartriangle ADB$. [b]L8. [/b]Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the circumcenter of triangle $ABC$, if it exists. Find the number of points $X$ so that $f^{2019}(X) = X$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1)); draw(Circle(origin, 1)^^A--D^^B--C); real r=0.05; pair M1=midpoint(M--D), M2=midpoint(M--A); draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D))); draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D))); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that (i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$; (ii) points $A,B,D,E$ lie on a circle in this order. Find the locus of the intersection points of lines $AD$ and $BE$.

Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area? $ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$

A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

Point $P$ lies on the median from vertex $C$ of a triangle $ABC$. Line $AP$ meets $BC$ at $X$, and line $BP$ meets $AC$ at $Y$ . Prove that if quadrilateral $ABXY$ is cyclic, then triangle $ABC$ is isosceles.

A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge? [i](3 points)[/i]