Found problems: 2265
1996 All-Russian Olympiad, 3
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
[i]N. Agakhanov, N. Tereshin[/i]
1968 IMO, 4
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
2008 National Olympiad First Round, 12
In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color?
$
\textbf{(A)}\ 154
\qquad\textbf{(B)}\ 203
\qquad\textbf{(C)}\ 210
\qquad\textbf{(D)}\ 240
\qquad\textbf{(E)}\ \text{None of the above}
$
2009 Sharygin Geometry Olympiad, 24
A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.
2018 Indonesia Juniors, day 2
P6. It is given the integer $Y$ with
$Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$
Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.)
P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....
P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$.
P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal.
(Image should be placed here, look at attachment.)
a) Determine the position of the number $2018$ based on its row, column, and diagonal.
b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$.
P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a [i]gadang[/i] number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a [i]gadang[/i] number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a [i]gadang[/i] number?
Champions Tournament Seniors - geometry, 2011.4
The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).
2025 Alborz Mathematical Olympiad, P3
Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points?
Proposed by Arvin Taheri
1990 IMO Longlists, 61
Prove that we can fill in the three dimensional space with regular tetrahedrons and regular octahedrons, all of which have the same edge-lengths. Also find the ratio of the number of the regular tetrahedrons used and the number of the regular octahedrons used.
2022 Denmark MO - Mohr Contest, 1
The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img]
a) What is the height of the liquid when it lies as shown in figure $2$?
b) What is the height of the liquid when it lies as shown in figure$ 3$?
2015 Sharygin Geometry Olympiad, 7
Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere.
Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are coplanar.
1962 IMO Shortlist, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
2007 Tournament Of Towns, 5
From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?
2007 All-Russian Olympiad, 1
Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd.
[i]A. Poliansky[/i]
2019 BMT Spring, 9
Let $ ABCD $ be a tetrahedron with $ \angle ABC = \angle ABD = \angle CBD = 90^\circ $ and $ AB = BC $. Let $ E, F, G $ be points on $ \overline{AD} $, $ BD $, and $ \overline{CD} $, respectively, such that each of the quadrilaterals $ AEFB $, $ BFGC $, and $ CGEA $ have an inscribed circle. Let $ r $ be the smallest real number such that $ \dfrac{[\triangle EFG]}{[\triangle ABC]} \leq r $ for all such configurations $ A, B, C, D, E, F, G $. If $ r $ can be expressed as $ \dfrac{\sqrt{a - b\sqrt{c}}}{d} $ where $ a, b, c, d $ are positive integers with $ \gcd(a, b) $ squarefree and $ c $ squarefree, find $ a + b + c + d $.
Note: Here, $ [P] $ denotes the area of polygon $ P $. (This wasn't in the original test; instead they used the notation $ \text{area}(P) $, which is clear but frankly cumbersome. :P)
2019 Jozsef Wildt International Math Competition, W. 58
In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$
2003 National High School Mathematics League, 11
Eight spheres with radius of $1$ are put into a circular column. There are two floors, and each sphere is tangent to adjacent four spheres, one of the bottom surfaces, and the flank. Then the height of the circular column is________.
1983 IMO Longlists, 67
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
1980 Czech And Slovak Olympiad IIIA, 6
Let $M$ be the set of five points in space, none of which four do not lie in a plane. Let $R$ be a set of seven planes with properties:
a) Each plane from the set $R$ contains at least one point of the set$ M$.
b) None of the points of the set M lie in the five planes of the set $R$.
Prove that there are also two distinct points $P$, $Q$, $ P \in M$, $Q \in M$, that the line $PQ$ is not the intersection of any two planes from the set $R$.
1991 Arnold's Trivium, 17
Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.
2008 Stanford Mathematics Tournament, 10
Six people play the following game: They have a cube, initially white. One by one, the players mark an $ X$ on a white face of the cube, and roll it like a die. The winner is the first person to roll an $ X$ (for example, player 1 wins with probability $ \frac {1}{6}$, while if none of players 1-5 win, player 6 will place an $ X$ on the last square and win for sure). What is the probability that the sixth player wins?
1982 USAMO, 5
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
1996 Tournament Of Towns, (518) 1
Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour?
(Mebius, Sharygin)
1981 Poland - Second Round, 6
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 $.
2020 Yasinsky Geometry Olympiad, 6
A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases.
(Alexander Shkolny)
2016 Sharygin Geometry Olympiad, P22
Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.