Found problems: 2265
2006 District Olympiad, 4
a) Prove that we can assign one of the numbers $1$ or $-1$ to the vertices of a cube such that the product of the numbers assigned to the vertices of any face is equal to $-1$.
b) Prove that for a hexagonal prism such a mapping is not possible.
2024 All-Russian Olympiad, 1
We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder?
[i]Proposed by A. Kuznetsov[/i]
1982 National High School Mathematics League, 9
In tetrahedron $SABC$, $\angle ASB=\frac{\pi}{2}, \angle ASC=\alpha(0<\alpha<\frac{\pi}{2}), \angle BSC=\beta(0<\beta<\frac{\pi}{2})$.
Let $\theta=A-SC-B$, prove that $\theta=-\arccos(\cot\alpha\cdot\cot\beta)$.
2007 Romania Team Selection Test, 4
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
1985 Bundeswettbewerb Mathematik, 2
The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$
2008 USAPhO, 2
A uniform pool ball of radius $r$ and mass $m$ begins at rest on a pool table. The ball is given a horizontal impulse $J$ of fixed magnitude at a distance $\beta r$ above its center, where $-1 \le \beta \le 1$. The coefficient of kinetic friction between the ball and the pool table is $\mu$. You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass $m$ and radius $r$ is $I_{cm} = \frac{2}{5}mr^2$.)
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(a) Find an expression for the final speed of the ball as a function of $J$, $m$, and $\beta$.
(b) For what value of $\beta$ does the ball immediately begin to roll without slipping, regardless of the value of $\mu$?
1979 IMO, 2
We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.
2005 India IMO Training Camp, 2
Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that
(i) $m^3 < a < b < c < (m+1)^3$;
(ii) $abc$ is the cube of an integer.
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$.
1964 Polish MO Finals, 6
Given is a pyramid $SABCD$ whose base is a convex quadrilateral $ ABCD $ with perpendicular diagonals $ AC $ and $ BD $, and the orthogonal projection of vertex $S$ onto the base is the point $0$ of the intersection of the diagonals of the base. Prove that the orthogonal projections of point $O$ onto the lateral faces of the pyramid lie on the circle.
2002 AMC 8, 22
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
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$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36$
1978 Romania Team Selection Test, 3
[b]a)[/b] Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $
[b]b)[/b] Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants.
[b]c)[/b] Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.
1973 Czech and Slovak Olympiad III A, 2
Given a tetrahedron $A_1A_2A_3A_4$, define an $A_1$-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex $A_1$ and the sphere are separated by the plane $A_2A_3A_4.$ Denote $\varrho_1,\ldots,\varrho_4$ of all four exspheres. Furthermore, denote $v_i, i=1,\ldots,4$ the distance of the vertex $A_i$ from the opposite face. Show that \[2\left(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+\frac{1}{v_4}\right)=\frac{1}{\varrho_1}+\frac{1}{\varrho_2}+\frac{1}{\varrho_3}+\frac{1}{\varrho_4}.\]
2007 F = Ma, 21
If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius?
$ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $
2019 Stanford Mathematics Tournament, 5
The bases of a right hexagonal prism are regular hexagons of side length $s > 0$, and the prism has height $h$. The prism contains some water, and when it is placed on a flat surface with a hexagonal face on the bottom, the water has depth $\frac{s\sqrt3}{4}$. The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute $\frac{h}{s}$.
2011-2012 SDML (High School), 14
How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$?
$\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$
2013 NZMOC Camp Selection Problems, 4
Let $C$ be a cube. By connecting the centres of the faces of $C$ with lines we form an octahedron $O$. By connecting the centers of each face of $O$ with lines we get a smaller cube $C'$. What is the ratio between the side length of $C$ and the side length of $C'$?
2005 International Zhautykov Olympiad, 3
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.
2018 Hong Kong TST, 2
For which natural number $n$ is it possible to place natural number from 1 to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way, that the sum of all the numbers, that surround each face is the same?
2014 IPhOO, 14
A super ball rolling on the floor enters a half circular track (radius $R$). The ball rolls without slipping around the track and leaves (velocity $v$) traveling horizontally in the opposite direction. Afterwards, it bounces on the floor. How far (horizontally) from the end of the track will the ball bounce for the second time? The ball’s surface has a theoretically infinite coefficient of static friction. It is a perfect sphere of uniform density. All collisions with the ground are perfectly elastic and theoretically instantaneous. Variations could involve the initial velocity being given before the ball enters the track or state that the normal force between the ball and the track right before leaving is zero (centripetal acceleration).
[i]Problem proposed by Brian Yue[/i]
2005 Romania National Olympiad, 2
The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.
2006 India IMO Training Camp, 2
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that
\[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\]
Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
1989 All Soviet Union Mathematical Olympiad, 508
A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).
2010 AIME Problems, 11
Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$. When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$, the volume of the resulting solid is $ \frac {m\pi}{n\sqrt {p}}$, where $ m$, $ n$, and $ p$ are positive integers, $ m$ and $ n$ are relatively prime, and $ p$ is not divisible by the square of any prime. Find $ m \plus{} n \plus{} p$.
1976 Polish MO Finals, 3
Prove that for each tetrahedron, the three products of pairs of opposite edges are sides of a triangle.