Found problems: 2265
2016 HMNT, 16-18
16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of
$$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$
18. An $8$ by $8$ grid of numbers obeys the following pattern:
1) The first row and first column consist of all $1$s.
2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$.
What is the number in the 8th row and 8th column?
1966 IMO Longlists, 21
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality
\[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\]
When does equality occur?
2000 Tournament Of Towns, 2
What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon?
(A Shapovalov)
1988 IMO Shortlist, 6
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
2003 AMC 10, 3
A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$ \textbf{(A)}\ 4.5 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$
2014 AIME Problems, 13
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5,$ no collection of $k$ pairs made by the child contains the shoes from exactly $k$ of the adults is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2017 Caucasus Mathematical Olympiad, 7
$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.
1978 IMO Longlists, 48
Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.
[i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.
1998 National High School Mathematics League, 12
In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.
Kvant 2020, M2598
Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$?
Mikhail Evdokimov
2013 IMC, 3
Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$.
[b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$.
[i]Proposed by Tomasz Tkocz, University of Warwick.[/i]
1997 Romania National Olympiad, 4
Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent.
a) Prove that $ABCD$ is a rhombus.
b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .
2021 Belarusian National Olympiad, 11.8
Watermelon(a sphere) with radius $R$ lies on a table. $n$ flies fly above the table, each at distance $\sqrt{2}R$ from the center of the watermelon. At some moment any fly couldn't see any of the other flies. (Flies can't see each other, if the segment connecting them intersects or touches watermelon).
Find the maximum possible value of $n$
2021 IOM, 6
Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$
2012 Kyoto University Entry Examination, 2
Given a regular tetrahedron $OABC$. Take points $P,\ Q,\ R$ on the sides $OA,\ OB,\ OC$ respectively. Note that $P,\ Q,\ R$ are different from the vertices of the tetrahedron $OABC$. If $\triangle{PQR}$ is an equilateral triangle, then prove that three sides $PQ,\ QR,\ RP$ are pararell to three sides $AB,\ BC,\ CA$ respectively.
30 points
1953 Moscow Mathematical Olympiad, 240
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
2009 AMC 8, 17
The positive integers $ x$ and $ y$ are the two smallest positive integers for which the product of $ 360$ and $ x$ is a square and the product of $ 360$ and $ y$ is a cube. What is the sum of $ x$ and $ y$?
$ \textbf{(A)}\ 80 \qquad
\textbf{(B)}\ 85 \qquad
\textbf{(C)}\ 115 \qquad
\textbf{(D)}\ 165 \qquad
\textbf{(E)}\ 610$
2004 Nicolae Păun, 3
[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point.
[i]Alexandru Sergiu Alamă[/i]
2012 Purple Comet Problems, 26
A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments as shown below. This cup has height $h$ and volume $V$. A second cup that is exactly the same shape as the first is held upright inside the fi rst cup so that its base is a distance of $\tfrac{h}2$ from the base of the fi rst cup. The volume of liquid that will t inside the fi rst cup and outside the second cup can be written $\tfrac{m}{n}\cdot V$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
pair s = (10,1);
draw(ellipse((0,0),4,1)^^ellipse((0,-6),2,.5));
fill((3,-6)--(-3,-6)--(0,-2.1)--cycle,white);
draw((4,0)--(2,-6)^^(-4,0)--(-2,-6));
draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5));
fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-2.1)--cycle,white);
draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6));
pair s = (10,-2);
draw(shift(s)*ellipse((0,0),4,1)^^shift(s)*ellipse((0,-6),2,.5));
fill(shift(s)*(3,-6)--shift(s)*(-3,-6)--shift(s)*(0,-4.1)--cycle,white);
draw(shift(s)*(4,0)--shift(s)*(2,-6)^^shift(s)*(-4,0)--shift(s)*(-2,-6));
//darn :([/asy]
1966 IMO Longlists, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
2009 Princeton University Math Competition, 4
Tetrahedron $ABCD$ has sides of lengths, in increasing order, $7, 13, 18, 27, 36, 41$. If $AB=41$, then what is the length of $CD$?
2014 Contests, 3
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
2012 Gulf Math Olympiad, 4
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.
[list](i) Prove that all the discs that he cuts have integral areas.
(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]
1995 Czech And Slovak Olympiad IIIA, 1
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.
1996 National High School Mathematics League, 11
Color the six faces of a cube in six given colors. Each face is colored in exactly one color. Two faces with a common edge is in different colors. Then the number of ways to color the cube is________.
Note: If we can make two cubes look the same by turning one of then, they are considered the same.