Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

Is it possible that the heights of a tetrahedron (that is, a triangular pyramid) would be equal to the numbers $1$, $2$, $3$ and $6$?

In $\triangle ABC$, $E\in AC$, $D\in AB$, $P=BE\cap CD$. Given that $S\triangle BPC=12$, while the areas of $\triangle BPD$, $\triangle CPE$ and quadrilateral $AEPD$ are all the same, which is $x$. Find the value of $x$.

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

In a convex quadrilateral $ ABCD, $ let $ A’,B’ $ be the orthogonal projections to $ CD $ of $ A, $ respectively, $ B. $ [b]a)[/b] Assuming that $ BB’\le AA’ $ and that the perimeter of $ ABCD $ is $ (AB+CD)\cdot BB’, $ is $ ABCD $ necessarily a trapezoid? [b]b)[/b] The same question with the addition that $ \angle BAD $ is obtuse.

Let $b$ be the length of the largest diagonal and $c$ be the length of the smallest diagonal of a regular nonagon with side length $a$. Which one of the followings is true? $ \textbf{(A)}\ b=\dfrac{a+c}2 \qquad\textbf{(B)}\ b=\sqrt {ac} \qquad\textbf{(C)}\ b^2=\dfrac{a^2+c^2}2 \\ \textbf{(D)}\ c=a+b \qquad\textbf{(E)}\ c^2=a^2+b^2 $

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

Let $ O $ be the circumcenter of $ ABC. $ The equalities $$ |OA+2OB|=|OB+2OC|=|OC+2OA| $$ hold. Prove that $ ABC $ is equilateral.

Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the side of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of the star. What is the difference between the maximum and minimum possible perimeter of $s$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(E)}\ \sqrt{5} $

Let $ABC$ be a triangle, and let $D$ and $D_1$ be points on segment $BC$ such that $BD = CD_1$. Construct point $E$ such that $EC\perp BC$ and $ED\perp AC$. Similarly, construct point $F$ such that $FB\perp BC$ and $FD\perp AB$. Prove that $EF\perp AD_1$.

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]

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$ (1) Find the condition for which $C_1$ is inscribed in $C_2$. (2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$. (3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$. 60 point

Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces. [i]Proposed by Kei Irei, Japan[/i]

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

Find the smallest positive integer whose cube ends in 888.

Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)

Given an isosceles triangle $ABC$ such that $|AB|=|AC|$ . Let $M$ be the midpoint of the segment $BC$ and let $P$ be a point other than $A$ such that $PA\parallel BC$. The points $X$ and $Y$ are located respectively on rays $PB$ and $PC$, so that the point $B$ is between $P$ and $X$, the point $C$ is between $P$ and $Y$ and $\angle PXM=\angle PYM$. Prove that the points $A,P,X$ and $Y$ are concyclic.

In $ \vartriangle ABC$, $\angle B = 46^o$ and $\angle C = 48^o$ . A circle is inscribed in $ \vartriangle ABC$ and the points of tangency are connected to form $PQR$. What is the measure of the largest angle in $\vartriangle P QR$?

In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$

Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.