Let $U$ be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices $O,A,B,C$. Let $V$ be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that $V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}$.

The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.

A regular hexagon and an equilateral triangle have equal perimeter. If the area of the triangle is $4\sqrt3$ square units, the area of the hexagon is (A): $5\sqrt3$, (B): $6\sqrt3$, (C): $7\sqrt3$, (D): $8\sqrt3$, (E): None of the above.

[b]p1.[/b] Denote $S_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}$. What is $144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})$? [b]p2.[/b] Let $A,B,C$ be three collinear points, with $AB = 4$, $BC = 8$, and $AC = 12$. Draw circles with diameters $AB$, $BC$, and $AC$. Find the radius of the two identical circles that will lie tangent to all three circles. [b]p3.[/b] Let $s(i)$ denote the number of $1$’s in the binary representation of $i$. What is $$\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right) \,\, mod \,\,629 ?$$ [b]p4.[/b] Parallelogram $ABCD$ has an area of $S$. Let $k = 42$. $E$ is drawn on AB such that $AE =\frac{AB}{k}$ . $F$ is drawn on $CD$ such that $CF = \frac{CD}{k}$ . $G$ is drawn on $BC$ such that $BG = \frac{BC}{k}$ . $H$ is drawn on $AD$ such that $DH = \frac{AD}{k}$ . Line $CE$ intersects $BH$ at $M$, and $DG$ at $N$. Line $AF$ intersects $DG$ at $P$, and $BH$ at $Q$. If $S_1$ is the area of quadrilateral $MNPQ$, find $\frac{S_1}{S}$. [b]p5.[/b] Let $\phi$ be the Euler totient function. What is the sum of all $n$ for which $\frac{n}{\phi(n)}$ is maximal for $1 \le n \le 500$? [b]p6.[/b] Link starts at the top left corner of an $12 \times 12$ grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly $6$ turns, in how many ways can he reach his destination? PS. You had better use hide for answers.

Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

Let $\odot O$ , $\odot I$ be the circumcircle and inscribed circles of triangle$ABC$ . Prove that : From every point $D$ on $\odot O$ ,we can construct a triangle $DEF$ such that $ABC$ and $DEF$ have the same circumcircle and inscribed circles

In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is $ \textbf{(A)}\ 1/2\qquad \textbf{(B)}\ 2/3\qquad \textbf{(C)}\ 3/4\qquad \textbf{(D)}\ 5/6\qquad \textbf{(E)}\ 4/3$ [asy] size(180); defaultpen(linewidth(.7pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label("$\textbf{A}$",foot((0,2),(0,3),(1,2)),SW); label("$\textbf{B}$",midpoint((0,1)--(1,2))); label("$\textbf{C}$",midpoint((-1,0)--(0,1))); label("$\textbf{D}$",midpoint((0,0)--(1,1))); label("$\textbf{E}$",midpoint((1,0)--(2,1)),NW); label("$\textbf{F}$",midpoint((0,-1)--(1,0)),NW); label("$\textbf{G}$",midpoint((0,-1)--(1,0)),2SE);[/asy]

Assume a triangle $ABC$ satisfies $|AB| = 1, |AC| = 2$ and $\angle ABC = \angle ACB + 90^\circ.$ What is the area of $ABC?$ \[ \mathrm a. ~ 6/7\qquad \mathrm b.~5/7\qquad \mathrm c. ~1/2 \qquad \mathrm d. ~4/5 \qquad \mathrm e. ~3/5 \]

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.

Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^{o}$.

In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.

Pete claims that he can draw $4$ segments of length $1$ and a circle of radius less than $\sqrt3 /3 $ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $4$ segments. Is Pete right?

In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png[/img]

Suppose $ABCD$ is an isosceles trapezoid in which $\overline{AB}\parallel\overline{CD}$. Two mutually externally tangent circles $\omega_1$ and $\omega_2$ are inscribed in $ABCD$ such that $\omega_1$ is tangent to $\overline{AB}$,$\overline{BC}$, and $\overline{CD}$ while $\omega_2$ is tangent to $\overline{AB}$, $\overline{DA}$, and $\overline{CD}$. Given that $AB=1$, $CD=6$, compute the radius of either circle.

Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.

Let $AB$ be an arbitrary chord of the circle $(O)$. Two circles $(X)$ and $(Y )$ are on the same side of the chord $AB$ such that they are both internally tangent to $(O)$ and they are tangent to $AB$ at $C,D$ respectively, $C$ is between $A$ and $D$. Let $H$ be the intersection of $XY$ and $AB, M$ the midpoint of arc $AB$ not containing $X$ and $Y$ . Let $HM$ meet $(O)$ again at $I$. Let $IX, IY$ intersect $AB$ again at $K, J$. Prove that the circumcircle of triangle $IKJ$ is tangent to $(O)$. Nguyễn Văn Linh

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Point $P$ is taken in the interior of the triangle $ABC$, so that \[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\] Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$. [i]Proposed by S. Berlov[/i]

Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.