Found problems: 2265
1985 Vietnam National Olympiad, 3
A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB \plus{}OC}{2}$, $ \frac{OC \plus{} OA}{2}$ and $ \frac{OA \plus{} OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.
2011 Indonesia MO, 5
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$.
1991 IMTS, 4
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that
\[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]
1966 IMO Shortlist, 20
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.
[b]a.)[/b] What is the volume of this polyhedron ?
[b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
1959 Polish MO Finals, 3
Given a pyramid with square base $ ABCD $ and vertex $ S $. Find the shortest path whose starting and ending point is the point $ S $ and which passes through all the vertices of the base.
1999 Gauss, 15
A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green?
$\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$
2009 AMC 10, 25
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of its opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
$ \textbf{(A)}\ \frac {1}{8}\qquad \textbf{(B)}\ \frac {3}{16}\qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {3}{8}\qquad \textbf{(E)}\ \frac {1}{2}$
2019 Yasinsky Geometry Olympiad, p2
The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD =
10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.
2023 Sharygin Geometry Olympiad, 24
A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.
1996 AMC 12/AHSME, 27
Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
balls?
$\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$
2008 USA Team Selection Test, 9
Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if
a) $ n$ is a positive integer not divisible by the square of a prime.
b) $ n$ is a positive integer not divisible by the cube of a prime.
2015 AIME Problems, 15
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.
[asy]import three; import solids;
size(8cm);
currentprojection=orthographic(-1,-5,3);
picture lpic, rpic;
size(lpic,5cm);
draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight);
draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight);
draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight);
draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8));
draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed);
draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed);
draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0));
draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8));
size(rpic,5cm);
draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight);
draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight);
draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed);
draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8));
draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8));
draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8));
draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8));
label(rpic,"$A$",(-3,3*sqrt(3),0),W);
label(rpic,"$B$",(-3,-3*sqrt(3),0),W);
add(lpic.fit(),(0,0));
add(rpic.fit(),(1,0));[/asy]
1993 National High School Mathematics League, 13
In triangular pyramid $S-ABC$, any two of $SA,SB,SC$ are perpendicular. $M$ is the centre of gravity of $\triangle ABC$. $D$ is the midpoint of $AB$, line $DP//SC$. Prove:
[b](a)[/b] $DP$ and $SM$ intersect.
[b](b)[/b] $DP\cap SM=D'$, then $D'$ is the center of circumsphere of $S-ABC$.
2002 German National Olympiad, 2
Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$.
a) Prove there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,15R$.
b) Does there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,2R$?
1992 IMO Longlists, 76
Given any triangle $ABC$ and any positive integer $n$, we say that $n$ is a [i]decomposable[/i] number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$. Determine the positive integers that are decomposable numbers for every triangle.
1994 All-Russian Olympiad Regional Round, 11.7
Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.
1998 Tournament Of Towns, 1
A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.
1976 IMO Longlists, 17
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.
2014 Flanders Math Olympiad, 1
(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals.
(b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that
$$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$
1997 Austrian-Polish Competition, 9
Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.
1987 IMO Shortlist, 10
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
1950 AMC 12/AHSME, 5
If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series:
$\textbf{(A)}\ 648 \qquad
\textbf{(B)}\ 832 \qquad
\textbf{(C)}\ 1168 \qquad
\textbf{(D)}\ 1944 \qquad
\textbf{(E)}\ \text{None of these}$
MMPC Part II 1958 - 95, 1967
[b]p1.[/b] Consider the system of simultaneous equations
$$(x+y)(x+z)=a^2$$
$$(x+y)(y+z)=b^2$$
$$(x+z)(y+z)=c^2$$
, where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$.
[b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ .
[img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img]
[b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer.
[b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img]
[b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2001 German National Olympiad, 6 (11)
In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$.
2019 Polish Junior MO First Round, 7
A cube $ABCDA'B'C'D'$ is given with an edge of length $2$ and vertices marked as in the figure. The point $K$ is center of the edge $AB$. The plane containing the points $B',D', K$ intersects the edge $AD$ at point $L$. Calculate the volume of the pyramid with apex $A$ and base the quadrilateral $D'B'KL$.
[img]https://cdn.artofproblemsolving.com/attachments/7/9/721989193ffd830fd7ad43bdde7e177c942c76.png[/img]