Given a real number $x,$ let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{1}, n_{2}, \ldots, n_{70}$ such that $k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor$ and $k$ divides $n_{i}$ for all $i$ such that $1 \leq i \leq 70.$ Find the maximum value of $\frac{n_{i}}{k}$ for $1\leq i \leq 70.$

Given an isosceles trapezoid $ABCD$ with $AB = 6$, $CD = 12$, and area $36$, find $BC$.

Let $ ABC$ a triangle. Let $D$ on $AB$ and $E$ on $AC$ such that $DE||BC$. Let line $DE$ intersect circumcircle of $ABC$ at two distinct points $F$ and $G$ so that line segments $BF$ and $CG$ intersect at P. Let circumcircle of $GDP$ and $FEP$ intersect again at $Q$. Prove that $A, P, Q$ are collinear.

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?

A polygon is [b]gridded[/b] if the internal angles of the polygon are either $90$ or $270$, it has integer side lengths and its sides don't intersect with each other. Prove that for all $n \ge 8$, it exist a gridded polygon with area $2n$ and perimeter $2n$.

Given an isosceles $ABC$, which has $2AC = AB + BC$. Denote $I$ the center of the inscribed circle, $K$ the midpoint of the arc $ABC$ of the circumscribed circle. Let $T$ be such a point on the line $AC$ that $\angle TIB = 90 {} ^ \circ$. Prove that the line $TB$ touches the circumscribed circle $\Delta KBI$. (Anton Trygub)

Let $\triangle ABC$ be an arbitary triangle, and construct $P,Q,R$ so that each of the angles marked is $30^\circ$. Prove that $\triangle PQR$ is an equilateral triangle. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair ext30(pair pt1, pair pt2) { pair r1 = pt1+rotate(-30)*(pt2-pt1), r2 = pt2+rotate(30)*(pt1-pt2); draw(anglemark(r1,pt1,pt2,25)); draw(anglemark(pt1,pt2,r2,25)); return intersectionpoints(pt1--r1, pt2--r2)[0]; } pair A = (0,0), B=(10,0), C=(3,7), P=ext30(B,C), Q=ext30(C,A), R=ext30(A,B); draw(A--B--C--A--R--B--P--C--Q--A); draw(P--Q--R--cycle, linetype("8 8")); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, NE); label("$Q$", Q, NW); label("$R$", R, S);[/asy]

Point $P$ lies inside of scalene triangle $ABC$ with incenter $I$ such that $:$ $$ 2\angle ABP = \angle BCA , 2\angle ACP = \angle CBA $$ Lines $PB$ and $PC$ intersect line $AI$ respectively at $B'$ and $C'$. Line through $B'$ parallel to $AB$ intersects $BI$ at $X$ and line through $C'$ parallel to $AC$ intersects $CI$ at $Y$. Prove that triangles $PXY$ and $ABC$ are similar.

Given an equilateral $\Delta ABC$, in which ${{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}}$ are the midpoint of the sides $ BC, \, \, AC, \, \, AB$ respectively. The line $l$ passes through the vertex $A$, we denote by $P, Q$ the projection of the points $B, C$ on the line $l$, respectively (the line $ l $ and the point $Q, \, \, A, \, \, P$ are located as shown in fig.). Denote by $T $ the intersection point of the lines ${{B} _ {1}} P$ and ${{C} _ {1}} Q$. Prove that the line ${{A} _ {1}} T$ is perpendicular to the line $l$. [img]https://cdn.artofproblemsolving.com/attachments/4/b/61f2f4ec9e6b290dfcd47e9351110bebd3bd43.png[/img] (Serdyuk Nazar)

In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.

A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]

Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

One convex quadrilateral is inside another. Can it turn out that the sum of the lengths of the diagonals of the outer quadrilateral is less than the sum of the lengths of the diagonals of the inner?

Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area. Prove that $S \le \frac{ (a + b)(c + d)}{4}$

Let $ABCD$ be a cyclic quadrilateral.$E$ is the midpoint of $(AC)$ and $F$ is the midpoint of $(BD)$ {$G$}=$AB\cap CD$ and {$H$}=$AD\cap BC$. a)Prove that the intersections of the angle bisector of $\angle{AHB}$ and the sides $AB$ and $CD$ and the intersections of the angle bisector of$\angle{AGD}$ with $BC$ and $AD$ are the verticles of a rhombus b)Prove that the center of this rhombus lies on $EF$

Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).

The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality \[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? $ \textbf{(A)}\ \frac {5\sqrt {2} \minus{} 7}{3}\qquad \textbf{(B)}\ \frac {10 \minus{} 7\sqrt {2}}{3}\qquad \textbf{(C)}\ \frac {3 \minus{} 2\sqrt {2}}{3}\qquad \textbf{(D)}\ \frac {8\sqrt {2} \minus{} 11}{3}\qquad \textbf{(E)}\ \frac {6 \minus{} 4\sqrt {2}}{3}$

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

Two circles are tangent externaly at a point $B$. A line tangent to one of the circles at a point $A$ intersects the other circle at points $C$ and $D$. Show that $A$ is equidistant to the lines $BC$ and $BD$.

A circle of radius $ 1$ is surrounded by $ 4$ circles of radius $ r$ as shown. What is $ r$? [asy]defaultpen(linewidth(.9pt)); real r = 1 + sqrt(2); pair A = dir(45)*(r + 1); pair B = dir(135)*(r + 1); pair C = dir(-135)*(r + 1); pair D = dir(-45)*(r + 1); draw(Circle(origin,1)); draw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r)); draw(A--(dir(45)*r + A)); draw(B--(dir(45)*r + B)); draw(C--(dir(45)*r + C)); draw(D--(dir(45)*r + D)); draw(origin--(dir(25))); label("$r$",midpoint(A--(dir(45)*r + A)), SE); label("$r$",midpoint(B--(dir(45)*r + B)), SE); label("$r$",midpoint(C--(dir(45)*r + C)), SE); label("$r$",midpoint(D--(dir(45)*r + D)), SE); label("$1$",origin,W);[/asy]$ \textbf{(A)}\ \sqrt {2}\qquad \textbf{(B)}\ 1 \plus{} \sqrt {2}\qquad \textbf{(C)}\ \sqrt {6}\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 2 \plus{} \sqrt {2}$

Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that: $$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$ Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas. [i]Petre Asaftei[/i]

Let $ABC$ be a triangle. Let $X$ be the point on side $AB$ such that $\angle{BXC} = 60^{\circ}$. Let $P$ be the point on segment $CX$ such that $BP\bot AC$. Given that $AB = 6, AC = 7,$ and $BP = 4,$ compute $CP$.

On top of a thin square cake are triangular chocolate chips which are mutually disjoint. Is it possible to cut the cake into convex polygonal pieces each containing exactly one chip?