Found problems: 2265
2014 Online Math Open Problems, 24
Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by $10^6$.
[i]Proposed by Robin Park[/i]
2011 Math Prize For Girls Problems, 7
If $z$ is a complex number such that
\[
z + z^{-1} = \sqrt{3},
\]
what is the value of
\[
z^{2010} + z^{-2010} \, ?
\]
1981 IMO Shortlist, 18
Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet.
1981 AMC 12/AHSME, 22
How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four?
$\text{(A)} \ 60 \qquad \text{(B)} \ 64 \qquad \text{(C)} \ 72 \qquad \text{(D)} \ 76 \qquad \text{(E)} \ 100$
2008 AIME Problems, 3
A block of cheese in the shape of a rectangular solid measures $ 10$ cm by $ 13$ cm by $ 14$ cm. Ten slices are cut from the cheese. Each slice has a width of $ 1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?
1997 Israel Grosman Mathematical Olympiad, 4
Prove that if two altitudes of a tetrahedron intersect, then so do the other two altitudes.
1994 National High School Mathematics League, 5
In regular $n$-regular pyramid, the range value of dihedral angle of two adjacent sides is
$\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)$
1983 Vietnam National Olympiad, 3
Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.
2015 BMT Spring, 16
Five points $A, B, C, D$, and $E$ in three-dimensional Euclidean space have the property that $AB = BC = CD = DE = EA = 1$ and $\angle ABC = \angle BCD =\angle CDE = \angle DEA = 90^o$ . Find all possible $\cos(\angle EAB)$.
2010 Brazil National Olympiad, 3
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?
Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
1965 Miklós Schweitzer, 5
Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.
Ukrainian TYM Qualifying - geometry, 2013.17
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
1970 IMO Shortlist, 3
In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?
2021 Girls in Math at Yale, 11
A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?
[i]Proposed by Deyuan Li and Andrew Milas[/i]
1958 Czech and Slovak Olympiad III A, 4
Consider positive numbers $d,v$ such that $d>v$. Moreover, consider two perpendicular skew lines $p,q$ of distance $v$ (that is direction vectors of both lines are orthogonal and $\min_{X\in p,Y\in q}XY = v$). Finally, consider all line segments $PQ$ such that $P\in p, Q\in q, PQ=d$.
a) Find the locus of all points $P$.
b) Find the locus of all midpoints of segments $PQ$.
1999 Romania National Olympiad, 2
On the sides $(AB)$, $(BC)$, $(CD)$ and $(DA)$ of the regular tetrahedron $ABCD$, one considers the points $M$, $N$, $P$, $Q$, respectively Prove that
$$MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.$$
2014 IPhOO, 15
The period of a given pendulum on a planet of radius $R$ is constant (unchanged) as we go from the surface of the planet down to radius $a$, where $R > a$. The planet has mass density evenly distributed at any radius $ r < a $. This density is $\rho_0$. Find the total mass of the planet. Express your answer in terms of $\rho_0$, $a$, $R$, the period of the pendulum, $T$, the length of the pendulum string, $L$, and other constants, as necessary.
[b]Warning[/b]: Your answer may contain some math. So be sure to input this correctly!
[i]Problem proposed by Trung Phan[/i]
2006 AMC 12/AHSME, 24
Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which
\[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34?
\]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$
1999 IMO, 1
A set $ S$ of points from the space will be called [b]completely symmetric[/b] if it has at least three elements and fulfills the condition that for every two distinct points $ A$ and $ B$ from $ S$, the perpendicular bisector plane of the segment $ AB$ is a plane of symmetry for $ S$. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.
2000 Saint Petersburg Mathematical Olympiad, 11.2
Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality:
$$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$
[I]Proposed by A. Khrabrov[/i]
1970 Spain Mathematical Olympiad, 1
A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined $30^o$ with respect to the vertical, we observe that when removing liquid so that the level falls $1$ cm, the weight of the contents decreases $40$ g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of $45^o$ with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.
2017 Yasinsky Geometry Olympiad, 5
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .
MIPT student olimpiad spring 2022, 2
Prove that every section of the cube $Q = {[-1,1]}^n \subset R^n$ linear k-dimensional subspace $L\subseteq R^n$ has a diameter of at least $2\sqrt k$.
1998 Vietnam National Olympiad, 2
Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.
2018 All-Russian Olympiad, 6
Three diagonals of a regular $n$-gon prism intersect at an interior point $O$. Show that $O$ is the center of the prism.
(The diagonal of the prism is a segment joining two vertices not lying on the same face of the prism.)