Let triangle $\vartriangle AEF$ be inscribed in a square $ABCD$ such that $E$ lies on $BC$ and $F$ lies on $CD$. If $\angle EAF = 45^o$ and $\angle BEA = 70^o$, compute $\angle CF E$.

$\vartriangle ABC$ is an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of $BC$ and $E$ be the point on AC such that $AE :CE = 5 : 3$. Let $X$ be the intersection of $BE$ and $AM$. Given that the area of $\vartriangle CM X$ is $15$, find the area of $\vartriangle ABC$.

Let $ABCD$ be a cyclic quadrilateral inscirbed in circle $O$. Let the tangent to $O$ at $A$ meet $BC$ at $S$, and the tangent to $O$ at $B$ meet $CD$ at $T$. Circle with $S$ as its center and passing $A$ meets $BC$ at $E$, and $AE$ meets $O$ again at $F(\ne A)$. The circle with $T$ as its center and passing $B$ meets $CD$ at $K$. Let $P = BK \cap AC$. Prove that $P,F,D$ are collinear if and only if $AB = AP$.

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.

Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$. [b](a)[/b] Prove that $r_{1}+r_{2}=2r$. [b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

Prove that for a triangle $ ABC $ with $ \angle BAC \ge 90^{\circ } , $ having circumradius $ R $ and inradius $ r, $ the following inequality holds: $$ R\sin A>2r $$ [i]Romeo Ilie[/i]

A cylindrical oil tank, lying horizontally, has an interior length of $ 10$ feet and an interior diameter of $ 6$ feet. If the rectangular surface of the oil has an area of $ 40$ square feet, the depth of the oil is: $ \textbf{(A)}\ \sqrt{5} \qquad \textbf{(B)}\ 2\sqrt{5} \qquad \textbf{(C)}\ 3\minus{}\sqrt{5} \qquad \textbf{(D)}\ 3\plus{}\sqrt{5} \\ \textbf{(E)}\ \text{either }3\minus{}\sqrt{5}\text{ or }3\plus{}\sqrt{5}$

Let $n$ be the answer to this problem. In acute triangle $ABC$, point $D$ is located on side $BC$ so that $\angle BAD = \angle DAC$ and point $E$ is located on $AC$ so that $BE \perp AC$. Segments $BE$ and $AD$ intersect at $X$ such that $\angle BXD = n^o$: Given that $\angle XBA = 16^o$, find the measure of $\angle BCA$.

Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

Let $AB$ be a segment of length $1.$ Let $\odot A, \odot B$ be circles with radius $\overline{AB}$ centered at $A, B.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote the intersection points of $\odot C$ and $\odot D$ by $E, F.$ Draw circles $\odot E, \odot F$ with radius $\overline{EF}$ and denote their intersection points $G, H.$ Compute the area of the pentagon $ACFHE.$

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$. [i](Proposed by Mykhailo Shtandenko)[/i]

There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?

Let $ABCDEF$ be a convex hexagon containing a point $P$ in its interior such that $PABC$ and $PDEF$ are congruent rectangles with $PA = BC = P D = EF$ (and $AB = PC = DE = PF$). Let $\ell$ be the line through the midpoint of $AF$ and the circumcentre of $PCD$. Prove that $\ell$ passes through $P$.

The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.

In a triangle $ABC$ with $AB=AC$, let $D$ be the midpoint of $[BC]$. A line passing through $D$ intersects $AB$ at $K$, $AC$ at $L$. A point $E$ on $[BC]$ different from $D$, and a point $P$ on $AE$ is taken such that $\angle KPL=90^\circ-\frac{1}{2}\angle KAL$ and $E$ lies between $A$ and $P$. The circumcircle of triangle $PDE$ intersects $PK$ at point $X$, $PL$ at point $Y$ for the second time. Lines $DX$ and $AB$ intersect at $M$, and lines $DY$ and $AC$ intersect at $N$. Prove that the points $P,M,A,N$ are concyclic.

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $£$, lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$. The ray $PK$ meets $\omega$ at point $Q$. Prove that the circumcircle of triangle $EQF$ touches $\omega$.

Inside a quadrilateral $ABCD$ there exists a point $P$ such that $AP$ is perpendicular to $AD$ and the line $BP$ is perpendicular to $DC$. Besides, $AB = 7$, $AP = 3$, $BP = 6$, $AD = 5 $ and $CD = 10$. Calculate the area of the triangle $ABC$.

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.