Found problems: 2265
2003 District Olympiad, 4
a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$.
b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular.
Valentin Vornicu
2009 AMC 8, 25
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cub is $\tfrac12$ foot from the top face. The second cut is $\tfrac13$ foot below the first cut, and the third cut is $\tfrac1{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
[asy]
import three;
real d=11/102;
defaultpen(fontsize(8));
defaultpen(linewidth(0.8));
currentprojection=orthographic(1,8/15,7/15);
draw(unitcube, white, thick(), nolight);
void f(real x) {
draw((0,1,x)--(1,1,x)--(1,0,x));
}
f(d);
f(1/6);
f(1/2);
label("A", (1,0,3/4), W);
label("B", (1,0,1/3), W);
label("C", (1,0,1/6-d/4), W);
label("D", (1,0,d/2), W);
label("1/2", (1,1,3/4), E);
label("1/3", (1,1,1/3), E);
label("1/17", (0,1,1/6-d/4), E);[/asy]
[asy]
import three;
real d=11/102;
defaultpen(fontsize(8));
defaultpen(linewidth(0.8));
currentprojection=orthographic(2,8/15,7/15);
int t=0;
void f(real x) {
path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle;
path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle;
path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle;
draw(surface(r), white, nolight);
draw(surface(f), white, nolight);
draw(surface(u), white, nolight);
draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x));
t=t+1;
}
f(d);
f(1/2);
f(1/3);
f(1/17);
label("D", (1/2, 1, 0), SE);
label("A", (1+1/2, 1, 0), SE);
label("B", (2+1/2, 1, 0), SE);
label("C", (3+1/2, 1, 0), SE);[/asy]
$\textbf{(A)}\:6\qquad
\textbf{(B)}\:7\qquad
\textbf{(C)}\:\frac{419}{51}\qquad
\textbf{(D)}\:\frac{158}{17}\qquad
\textbf{(E)}\:11$
1995 Vietnam National Olympiad, 1
Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?
2016 HMNT, 9
A cylinder with radius $15$ and height $16$ is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$?
2007 Junior Balkan Team Selection Tests - Romania, 1
Find the positive integers $n$ with $n \geq 4$ such that $[\sqrt{n}]+1$ divides $n-1$ and $[\sqrt{n}]-1$ divides $n+1$.
[hide="Remark"]This problem can be solved in a similar way with the one given at [url=http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006]Cono Sur Olympiad 2006[/url], problem 5.[/hide]
2006 CentroAmerican, 1
For $0 \leq d \leq 9$, we define the numbers \[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]Find the last digit of the number \[S_{0}+S_{1}+\cdots+S_{9}.\]
2013 Federal Competition For Advanced Students, Part 2, 6
Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$.
Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.
2025 China Team Selection Test, 4
Recall that a plane divides $\mathbb{R}^3$ into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube $ABCD-A_1B_1C_1D_1$ lie on, and the four planes that the tetrahedron $ACB_1D_1$ lie on. How many regions do these ten planes split the space into?
2001 Romania National Olympiad, 2
In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality
\[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]
Estonia Open Senior - geometry, 2003.1.2
Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.
2012 AIME Problems, 5
In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5));
[/asy]
2007 National Olympiad First Round, 10
How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the above}
$
1978 IMO Longlists, 46
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
2011 Sharygin Geometry Olympiad, 13
a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex).
b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).