Found problems: 2265
1953 Czech and Slovak Olympiad III A, 4
Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.
2008 ITest, 89
Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points, $A$ and $B$, such that $AB=42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere.
1972 IMO Longlists, 13
Given a sphere $K$, determine the set of all points $A$ that are vertices of some parallelograms $ABCD$ that satisfy $AC \le BD$ and whose entire diagonal $BD$ is contained in $K$.
1973 IMO, 2
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
2020 Polish Junior MO First Round, 7.
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.
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]
2015 Math Prize for Girls Olympiad, 2
A tetrahedron $T$ is inside a cube $C$. Prove that the volume of $T$ is at most one-third the volume of $C$.
Estonia Open Senior - geometry, 1995.1.3
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .
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.)
2014 Math Prize For Girls Problems, 10
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?
2004 All-Russian Olympiad, 1
Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?
1935 Moscow Mathematical Olympiad, 006
The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.
1997 Tournament Of Towns, (536) 1
A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube.
(V Proizvolov)
1965 Kurschak Competition, 3
A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces.
[img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]
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?
1990 Tournament Of Towns, (255) 3
(a) Some vertices of a dodecahedron are to be marked so that each face contains a marked vertex. What is the smallest number of marked vertices for which this is possible?
(b) Answer the same question, but for an icosahedron.
(G. Galperin, Moscow)
(Recall that a dodecahedron has $12$ pentagonal faces which meet in threes at each vertex, while an icosahedron has $20$ triangular faces which meet in fives at each vertex.)
1987 AIME Problems, 3
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
1990 AMC 12/AHSME, 10
An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\textbf{(A) }328\qquad
\textbf{(B) }329\qquad
\textbf{(C) }330\qquad
\textbf{(D) }331\qquad
\textbf{(E) }332\qquad$
1983 National High School Mathematics League, 4
In a tetrahedron, lengths of six edges are $2,3,3,4,5,5$. Find its largest volume.
1940 Moscow Mathematical Olympiad, 066
* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
1980 AMC 12/AHSME, 27
The sum $\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ equals
$\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)} \ \sqrt[3]{2} \qquad \text{(E)} \ \text{none of these}$
V Soros Olympiad 1998 - 99 (Russia), 11.5
It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane ($14$ values in total) take two different values. The smallest is $1$. What can the edge of a cube be equal to?
1997 Vietnam Team Selection Test, 1
Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying
\[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
\]
and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
2004 Pre-Preparation Course Examination, 6
Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds.
a) Prove that:
\[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)}
\]
b) Prove that
\[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1)
\]
c) Prove the statement of problem.
d) Prove Van der Waerden's Theorem.
1960 Poland - Second Round, 6
Calculate the volume of the tetrahedron $ ABCD $ given the edges $ AB = b $, $ AC = c $, $ AD = d $ and the angles $ \measuredangle CAD = \beta $, $ \measuredangle DAB = \gamma $ and $ \measuredangle BAC = \delta$.