Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

In triangle $ABC$, $\angle BAC=60^o$/ Let $\omega$ be a circle tangent to segment $AB$ at point $D$ and segment $AC$ at point $E$. Suppose $\omega$ intersects segment $BC$ at points $F$ and $G$ such that$ F$ lies in between $B$ and $G$. Given that $AD = FG = 4$ and $BF = \frac12$ , find the length of $CG$.

Given is the triangle $ABC$, with $| AB | + | AC | = 3 \cdot | BC | $. Let's denote $D, E$ also points that $BCDA$ and $CBEA$ are parallelograms. On the sides $AC$ and $AB$ sides, $F$ and $G$ are selected respectively so that $| AF | = | AG | = | BC |$. Prove that the lines $DF$ and $EG$ intersect at the line segment $BC$

Inside the inscribed quadrilateral $ABCD$ are marked points $P$ and $Q$, such that $\angle PDC + \angle PCB,$ $\angle PAB + \angle PBC,$ $\angle QCD + \angle QDA$ and $\angle QBA + \angle QAD$ are all equal to $90^\circ$. Prove that the line $PQ$ has equal angles with lines $AD$ and $BC$. [i]A. Pastor[/i]

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

The vertices of the convex quadrilateral $ABCD$ and the intersection point $S$ of its diagonals are integer points in the plane. Let $P$ be the area of $ABCD$ and $P_1$ the area of triangle $ABS$. Prove that \[\sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2\]

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear. (Bulgaria)

Let $ABCD$ be a convex quadrilateral with non perpendicular diagonals and with the sides $AB$ and $CD$ non parallel . Denote by $O$ the intersection of the diagonals , $H_1$ the orthocenter of the triangle $AOB$ and $H_2$ the orthocenter of the triangle $COD$ . Also denote with $M$ the midpoint of the side $AB$ and with $N$ the midpoint of the side $CD$ . Prove that $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

Two circles $(O_1,R_1)$,$(O_2,R_2)$ lie each external to the other. Find : a) the minimum length of the segment connecting points of the circles b) the max length of the segment connecting points of the circles

Consider $n \ge 2$ pairwise disjoint disks $D_1,D_2,\ldots,D_n$ on the Euclidean plane. For each $k=1,2,\ldots,n$, denote by $f_k$ the inversion with respect to the boundary circle of $D_k$. (Here, $f_k$ is defined at every point of the plane, except for the center of $D_k$.) How many fixed points can the transformation $f_n\circ f_{n-1}\circ\ldots\circ f_1$ have, if it is defined on the largest possible subset of the plane?

In the plane there is a square $ Q $ and a point $ P $. The point $ K $ runs through the perimeter of the square $ Q $. Find the locus of the vertex $ M $ of the equilateral triangle $ KPM $.

The incircle $\omega$ of triangle $ABC$, right angled at $C$, touches the circumcircle of its medial triangle at point $F$. Let $OE$ be the tangent to $\omega$ from the midpoint $O$ of the hypotenuse $AB$, distinct from $AB$. Prove that $CE = CF$.

Three circles lie inside a square. Each of them touches externally two remaining circles. Also each circle touches two sides of the square. Prove that two of these circles are congruent.

On the extension of the angle bisector $AL$ of the triangle $ABC$, a point $P$ is placed such that $P L = AL$. Prove that the perimeter of triangle $PBC$ does not exceed the perimeter of triangle $ABC$.

[b]p5.[/b] Find the number of three-digit integers that contain at least one $0$ or $5$. The leading digit of the three-digit integer cannot be zero. [b]p6.[/b] What is the sum of the solutions to $\frac{x+8}{5x+7} =\frac{x+8}{7x+5}$ [b]p7.[/b] Let $BC$ be a diameter of a circle with center $O$ and radius $4$. Point $A$ is on the circle such that $\angle AOB = 45^o$. Point $D$ is on the circle such that line segment$ OD$ intersects line segment $AC$ at $E$ and $OD$ bisects $\angle AOC$. Compute the area of $ADE$, which is enclosed by line segments $AE$ and $ED$ and minor arc $AD$. [b]p8. [/b] William is a bacteria farmer. He would like to give his fiance$ 2021$ bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those $2021$ bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals? PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel. (Moldova)

Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $P,Q,R$ be points on the sides $AB, BC, CA$ of the triangle such that $CQPR$ is a parallelogram. Show that the reflection of $P$ over $QR$ lies on the circumcircle of $ABC$.

Consider a parallelogram $ABCD$. a) Prove that if the incenter of the triangle $ABC$ is located on the diagonal $BD$, then the parallelogram $ABCD$ is a rhombus. b) Is the parallelogram $ABCD$ a rhombus whenever the circumcenter of the triangle $ABC$ is located on the diagonal $BD$?

There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$. A side of triangle $T_k$ is equal to $\frac{1}{k}$, $k = 1,2, . . . ,1979$. At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don’t intersect (points of contact are allowed)?

Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum $$ \frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$ has a constant value, i.e. independent of the position of the plane $ MNP $.

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)? (A Andzans)

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]

Let $ABC$ be a triangle with sides $BC = 13$, $CA = 14$ and $AB = 15$. We denote by $I$ the intersection point of the angle bisectors and $M$ to the midpoint of $AB$. The line $IM$ cuts at $P$ at the altitude drawn from $C$. Find the length of $CP$.