Found problems: 2265
2012 Kyoto University Entry Examination, 2
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
1983 Spain Mathematical Olympiad, 1
While Theophrastus was talking to Aristotle about the classification of plants, had a dog tied to a perfectly smooth cylindrical column of radius $r$, with a very fine rope that wrapped around the column and with a loop. The dog had the extreme free from the rope around his neck. In trying to reach Theophrastus, he put the rope tight and it broke. Find out how far from the column the knot was in the time to break the rope.
[hide=original wording]Mientras Teofrasto hablaba con Arist´oteles sobre la clasificaci´on de las plantas, ten´ıa un perro atado a una columna cil´ındrica perfectamente lisa de radio r, con una cuerda muy fina que envolv´ıa la columna y con un lazo. El perro ten´ıa el extremo libre de la cuerda cogido a su cuello. Al intentar alcanzar a Teofrasto, puso la cuerda tirante y ´esta se rompi´o. Averiguar a qu´e distancia de la columna estaba el nudo en el momento de romperse la cuerda.[/hide]
2017 Israel Oral Olympiad, 4
What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?
1980 Bulgaria National Olympiad, Problem 2
(a) Prove that the area of a given convex quadrilateral is at least twice the area of an arbitrary convex quadrilateral inscribed in it whose sides are parallel to the diagonals of the original one.
(b) A tetrahedron with surface area $S$ is intersected by a plane perpendicular to two opposite edges. If the area of the cross-section is $Q$, prove that $S>4Q$.
1984 Bulgaria National Olympiad, Problem 6
Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.
1962 Polish MO Finals, 6
Given three lines $ a $, $ b $, $ c $ pairwise skew. Is it possible to construct a parallelepiped whose edges lie on the lines $ a $, $ b $, $ c $?
2009 Romanian Masters In Mathematics, 3
Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that
\[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3,
\]
denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel.
[i]Nikolai Ivanov Beluhov, Bulgaria[/i]
1992 National High School Mathematics League, 3
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
$\text{(A)}2<\lambda\leq4\qquad\text{(B)}3<\lambda<4\qquad\text{(C)}2.5<\lambda\leq4.5\qquad\text{(D)}3.5<\lambda<5.5$
2009 Purple Comet Problems, 24
A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$.
[asy]
size(150);
real t=0.12;
void ball(pair x, real r, real h, bool ww=true)
{
pair xx=yscale(t)*x+(0,h);
path P=circle(xx,r);
unfill(P);
draw(P);
if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r));
}
pair X=(0,0);
real H=17, h=5, R=h/2;
draw(H*dir(120)--(0,0)--H*dir(60));
draw(ellipse((0,0.87*H),H/2,t*H/2));
pair Y=(R,h+2*R),C=(0,h);
real r;
for(int k=0;k<20;++k)
{
r=-(dir(30)*Y).x;
Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3;
}
ball(Y.x*dir(90),r,Y.y,false);
ball(X,R,h);
ball(Y.x*dir(-30),r,Y.y);
ball(Y.x*dir(210),r,Y.y);[/asy]
Indonesia Regional MO OSP SMA - geometry, 2019.1
Given cube $ ABCD.EFGH $ with $ AB = 4 $ and $ P $ midpoint of the side $ EFGH $. If $ M $ is the midpoint of $ PH $, find the length of segment $ AM $.
2008 Baltic Way, 14
Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?
2006 China Second Round Olympiad, 4
Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is
${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $
2009 Indonesia TST, 3
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.
1913 Eotvos Mathematical Competition, 2
Let $O$ and $O'$ designate two dìagonally opposite vertices of a cube. Bisect those edges of the cube that contain neither of the points $O$ and $O'$. Prove that these midpoints of edges lie in a plane and form the vertices of a regular hexagon
1977 Bulgaria National Olympiad, Problem 2
In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron.
[i]N. Nenov, N. Hadzhiivanov[/i]
1997 All-Russian Olympiad, 4
An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color.
[i]M. Smurov[/i]
1997 Estonia National Olympiad, 3
A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.
1989 IMO Longlists, 74
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
2015 Caucasus Mathematical Olympiad, 4
The midpoint of the edge $SA$ of the triangular pyramid of $SABC$ has equal distances from all the vertices of the pyramid. Let $SH$ be the height of the pyramid. Prove that $BA^2 + BH^2 = C A^2 + CH^2$.
2005 Iran MO (3rd Round), 1
We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$.
a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN.
b) The circle is not CN.
Which one of these sets are CN?
1) $A=\{x\in\mathbb R^3| |x|=1\}$
2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$
3) Graph of the function $f:[0,1]\to \mathbb R$ defined by
\[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]
1997 AMC 8, 17
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("$x$",(2.75,3.5),NNE);
label("$y$",(4.125,1.5),NNE);
[/asy]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$
1971 Bulgaria National Olympiad, Problem 6
In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that
(a) $m^2+n^2+p^2\ge18r^2$;
(b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.
Ukrainian TYM Qualifying - geometry, IV.11
In the tetrahedron $ABCD$, the point $E$ is the projection of the point $D$ on the plane $(ABC)$. Prove that the following statements are equivalent:
a) $C = E$ or $CE \parallel AB$
b) For each point M belonging to the segment $CD$, the following equation is satisfied
$$S^2_{\vartriangle ABM}= \frac{CM^2}{CD^2}\cdot S^2_{\vartriangle ABD}+\left(1- \frac{CM^2}{CD^2} \right)S^2_{\vartriangle ABC}$$
where $S_{\vartriangle XYZ}$ means the area of triangle $XYZ$.
2015 CHMMC (Fall), 6
The icosahedron is a convex, regular polyhedron consisting of $20$ equilateral triangle for faces. A particular icosahedron given to you has labels on each of its vertices, edges, and faces. Each minute, you uniformly at random pick one of the labels on the icosahedron. If the label is on a vertex, you remove it. If the label is on an edge, you delete the label on the edge along with any labels still on the vertices of that edge. If the label is on a face, you delete the label on the face along with any labels on the edges and vertices which make up that face. What is the expected number of minutes that pass before you have removed all labels from the icosahedron?
1962 Vietnam National Olympiad, 3
Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC \equal{} AD \equal{} BC \equal{} BD$?