Found problems: 2265
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ .
Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners.
Prove that Germán can always achieve his goal.
The sum of the base-$ 10$ logarithms of the divisors of $ 10^n$ is $ 792$. What is $ n$?
$ \textbf{(A)}\ 11\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 13\qquad
\textbf{(D)}\ 14\qquad
\textbf{(E)}\ 15$
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$.
[i](B. Dejean, 6 points)[/i]
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
$\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$
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]
$\textbf{(A)}\:6\qquad
\textbf{(B)}\:7\qquad
\textbf{(C)}\:\frac{419}{51}\qquad
\textbf{(D)}\:\frac{158}{17}\qquad
\textbf{(E)}\:11$
A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
A cylinder with radius $5$ and height $1$ is rolling on the (unslanted) floor. Inside the cylinder, there is water that has constant height $\frac{15}{2}$ as the cylinder rolls on the floor. What is the volume of the water?
Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
[i]A. Chukhnov[/i]
Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.
A real number $f(X)\neq 0$ is assigned to each point $X$ in the space.
It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have :
\[ f(O)=f(A)f(B)f(C)f(D). \]
Prove that $f(X)=1$ for all points $X$.
[i]Proposed by Aleksandar Ivanov[/i]
Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.
In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that
(a) $m^2+n^2+p^2\ge18r^2$;
(b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.
Let $ABCD$ and $ABEF$ be two squares situated in two perpendicular planes and let $O$ be the intersection of the lines $AE$ and $BF$. If $AB=4$ compute:
a) the distance from $B$ to the line of intersection between the planes $(DOC)$ and $(DAF)$;
b) the distance between the lines $AC$ and $BF$.
A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.
(a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?
(b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.
Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((0,10)--(4,13)--(14,13)--(10,10));
draw((10,0)--(14,3)--(14,13));
draw((0,0)--(4,3)--(4,13), dashed);
draw((4,3)--(14,3), dashed);
dot((0,0));
dot((0,10));
dot((10,10));
dot((10,0));
dot((4,3));
dot((14,3));
dot((14,13));
dot((4,13));
dot((14,8));
dot((5,0));
label("A", (0,0), SW);
label("B", (10,0), S);
label("C", (14,3), E);
label("D", (4,3), NW);
label("E", (0,10), W);
label("F", (10,10), SE);
label("G", (14,13), E);
label("H", (4,13), NW);
label("M", (5,0), S);
label("N", (14,8), E);
[/asy]
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that
$$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$
Let $S$ be a set, such that for every positive integer $n$, we have $|S\cap T|=1$, where $T=\{n,2n,3n\}$. Prove that if $2\in S$, then $13824\notin S$.
The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is
[asy]
unitsize(36);
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3));
draw((0,3)--(2.2,4.4)--(5.2,4.4));
fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black);
draw((1,3)--(3.2,4.4));
draw((2,3)--(4.2,4.4));
draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666));
draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333));
fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black);
fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black);
fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black);
fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black);
fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black);
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 22 \qquad \text{(E)}\ 24$
Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).