Find all natural numbers $n$ for which there exists a convex polyhedron with $n$ edges, with exactly one vertex having four edges and all other vertices having $3$ edges.

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

Given a plane $P$ and two fixed points $A$ and $B$ that do not lie in this plane. Denote two points $A'$ and $B'$ on plane $P$ and $M ,N$ the midpoints of the segments $AA'$, $BB'$. a) Determine the locus of the midpoint of the segment MN if the points are $A'$ and $B'$ move arbitrarily in plane $P$. b) A circle $O$ is considered in the plane $P$. Determine the locus $L$ of the midpoint of the segment $MN$ if the points $A'$ and $B'$ lie on the circle $O$ or inside it . c) $A'$ is assumed to be fixed on the circle $O$ or inside it and $B'$ is assumed to be movable inside it , except for $O$. Determine the locus of the point $B'$ such the above certain locus $L$ remains the same . Note: For b) and c) the following cases should be considered: 1. $A'$ and $B'$ are different, 2. $A'$ and $B'$ coincide.

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.

Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?

Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.

In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

Consider the pyramid $ VABCD, $ where $ V $ is the top and $ ABCD $ is a rectangular base. If $ \angle BVD = \angle AVC, $ then prove that the triangles $ VAC $ and $ VBD $ share the same perimeter and area.

The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. [i]Jordan Tabov[/i]

A real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.

[b]p1.[/b] Consider the system of simultaneous equations $$(x+y)(x+z)=a^2$$ $$(x+y)(y+z)=b^2$$ $$(x+z)(y+z)=c^2$$ , where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$. [b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ . [img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img] [b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer. [b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. [img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img] [b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]

Two points $ P$ and $ Q$ lie in the interior of a regular tetrahedron $ ABCD$. Prove that angle $ PAQ < 60^\circ$.

A pentagon $ABCDE$ inscribed in a circle for which $BC<CD$ and $AB<DE$ is the base of a pyramid with vertex $S$. If $AS$ is the longest edge starting from $S$, prove that $BS>CS$.

Let $n$ be positive integers. Denote the graph of $y=\sqrt{x}$ by $C,$ and the line passing through two points $(n,\ \sqrt{n})$ and $(n+1,\ \sqrt{n+1})$ by $l.$ Let $V$ be the volume of the solid obtained by revolving the region bounded by $C$ and $l$ around the $x$ axis.Find the positive numbers $a,\ b$ such that $\lim_{n\to\infty}n^{a}V=b.$

Let $ ABCDA'B'C'D'$ be a cube. On the sides $ (A'D')$, $ (A'B')$ and $ (A'A)$ we consider the points $ M_1$, $ N_1$ and $ P_1$ respectively. On the sides $ (CB)$, $ (CD)$ and $ (CC')$ we consider the points $ M_2$, $ N_2$ and $ P_2$ respectively. Let $ d_1$ be the distance between the lines $ M_1N_1$ and $ M_2N_2$, $ d_2$ be the distance between the lines $ N_1P_1$ and $ N_2P_2$, and $ d_3$ be the distance between the lines $ P_1M_1$ and $ P_2M_2$. Suppose that the distances $ d_1$, $ d_2$ and $ d_3$ are pairwise distinct. Prove that the lines $ M_1M_2$, $ N_1N_2$ and $ P_1P_2$ are concurrent.

P6. Determine all integer pairs $(x, y)$ satisfying the following system of equations. \[ \begin{cases} x + y - 6 &= \sqrt{2x + y + 1} \\ x^2 - x &= 3y + 5 \end{cases} \] P7. Determine the sum of all (positive) integers $n \leq 2019$ such that $1^2 + 2^2 + 3^2 + \cdots + n^2$ is an odd number and $1^1 + 2^2 + 3^3 + \cdots + n^n$ is also an odd number. P8. Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is $a^3\sqrt{2}$ cm$^3$, determine the volume of the largest sphere that can be fit inside such "8-plane" (octahedron). P9. Six-digit numbers $\overline{ABCDEF}$ with distinct digits are arranged from the digits 1, 2, 3, 4, 5, 6, 7, 8 with the rule that the sum of the first three numbers and the sum of the last three numbers are the same. Determine the probability that such arranged number has the property that either the first or last three digits (might be both) form an arithmetic sequence or a geometric sequence. [hide=Remarks (Answer spoiled)]It's a bit ambiguous whether the first or last three digits mentioned should be in that order, or not. If it should be in that order, the answer to this problem would be $\frac{1}{9}$, whereas if not, it would be $\frac{1}{3}$. Some of us agree that the correct interpretation should be the latter (which means that it's not in order) and the answer should be $\frac{1}{3}$. However since this is an essay problem, your interpretation can be written in your solution as well and it's left to the judges' discretion to accept your interpretation, or not. This problem is very bashy.[/hide] P10. $X_n$ denotes the number which is arranged by the digit $X$ written (concatenated) $n$ times. As an example, $2_{(3)} = 222$ and $5_{(2)} = 55$. For $A, B, C \in \{1, 2, \ldots, 9\}$ and $1 \leq n \leq 2019$, determine the number of ordered quadruples $(A, B, C, n)$ satisfying: \[ A_{(2n)} = 2 \left ( B_{(n)} \right ) + \left ( C_{(n)} \right )^2. \]

Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base.

Let $ABCD$ be a regular tetrahedron and let $E$ be a point inside the face $ABC$. Denote by $s$ the sum of the distances from $E$ to the faces $DAB$, $DBC$, $DCA$, and by $S$ the sum of the distances from $E$ to the edges $AB$, $BC$, $CA$. Then $\dfrac sS$ equals $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\dfrac{2\sqrt2}3\qquad\textbf{(C) }\dfrac{\sqrt6}2\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); label("$15$",(1.5,1.2),N); label("$11$",(4,2.3),N); label("$14$",(2.5,3.7),N);[/asy] $ \text{(A)}\ 75\qquad\text{(B)}\ 76\qquad\text{(C)}\ 78\qquad\text{(D)}\ 80\qquad\text{(E)}\ 81 $