The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("$1$",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("$2$",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("$3$",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("$4$",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("$5$",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("$6$",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("$7$",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("$8$",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("$9$",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle; (2) the origin ($0,0$) is the incenter of the triangle.

Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$

Construct a point $Q$ in triangle $ABC$ such that at least two of the segments $CQ, BQ, AQ$, divide the inscribed circle in half. For which triangles is this possible?

trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle with the two semicircles on the two parallel sides of the trapezoid facing outside the trapezoid and the other two semicircles facing inside the trapezoid as shown. The region bounded by these four semicircles has area $m + n\pi$, where m and n are positive integers. Find $m + n$. [img]https://3.bp.blogspot.com/-s8BoUPKVUQk/XoEaIYvaz4I/AAAAAAAALl0/ML0klwHogGYWkNhY6maDdI93_GkfL_eyQCK4BGAYYCw/s200/2018%2Bps%2Bhs9.png[/img]

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively. If $AC,BD,EG,FH$ concur at a point $O,$ prove that $ABCD$ is a parallelogram.

Areas of four surfaces of a tetrahedron are $S_1,S_2,S_3,S_4$. And the largest one of them is $S$. $\lambda=\frac{S_1+S_2+S_3+S_4}{S}$, then $\lambda$ always satisfies $\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$. Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$ (V. Prasolov )

Let $w$ be a circle of diameter $5$. Four lines were drawn dividing $w$ into $5$ "strips", each of width $1$. The strips were colored orange and purple alternatingly, as depicted. Which area is larger: the orange or the purple?

The surface areas of the bases of a given truncated triangular pyramid are equal to $ B_1 $ and $ B_2 $. This pyramid can be cut with a plane parallel to the bases so that a sphere can be inscribed in each of the obtained parts. Prove that the lateral surface area of the given pyramid is $ (\sqrt{B_1} + \sqrt{B_2})(\sqrt[4]{B_1} + \sqrt[4]{B_2})^2 $.

The circle $\omega_1$ passes through the center $O$ of the circle $\omega_2$ and meets it at points $A$ and $B$. The circle $\omega_3$ centered at $A$ with radius $AB$ meets $\omega_1$ and $\omega_2$ at points $C$ and $D$ (distinct from $B$). Prove that $C, O, D$ are collinear.

Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$

Let $f$ be a map of the plane into itself with the property that if $d(A,B)=1$, then $d(f(A),f(B))=1$, where $d(X,Y)$ denotes the distance between points $X$ and $Y$. Prove that for any positive integer $n$, $d(A,B)=n$ implies $d(f(A),f(B))=n$.

Let us call a convex hexagon $ABCDEF$ [i]boring [/i] if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$. a) Is every cyclic hexagon boring? b) Is every boring hexagon cyclic?

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]

Let $ABC$ be a triangle with $AB < AC$. Let $D$ be the intersection point of the internal bisector of angle $BAC$ and the circumcircle of $ABC$. Let $Z$ be the intersection point of the perpendicular bisector of $AC$ with the external bisector of angle $\angle{BAC}$. Prove that the midpoint of the segment $AB$ lies on the circumcircle of triangle $ADZ$. [i]Olimpiada de Matemáticas, Nicaragua[/i]

In isosceles $\triangle ABC$, $AB=AC$, points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$, $DF\parallel AC$. The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$. Let $DE$ meet $\omega_2$ at $K\neq E$. Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$. Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$, with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$. Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$, the triangle $MNP$ is right angled at $N$.

The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.

Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular. (a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$; (b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$. [i]Mircea Fianu[/i]