Found problems: 2265
2003 AMC 10, 9
Find the value of $ x$ that satisfies the equation
\[ 25^{\minus{}2}\equal{}\frac{5^{48/x}}{5^{26/x}\cdot25^{17/x}}.
\]$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9$
2018 AMC 10, 10
In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2.$ Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?
[asy]
size(250);
defaultpen(fontsize(10pt));
pair A =origin;
pair B = (4.75,0);
pair E1=(0,3);
pair F = (4.75,3);
pair G = (5.95,4.2);
pair C = (5.95,1.2);
pair D = (1.2,1.2);
pair H= (1.2,4.2);
pair M = ((4.75+5.95)/2,3.6);
draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H);
draw(B--C);
draw(F--G);
draw(A--D--H--C--D,dashed);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E1,W);
label("$F$",F,SW);
label("$G$",G,NE);
label("$H$",H,NW);
label("$M$",M,N);
dot(A);
dot(B);
dot(E1);
dot(F);
dot(G);
dot(C);
dot(D);
dot(H);
dot(M);
label("3",A/2+B/2,S);
label("2",C/2+G/2,E);
label("1",C/2+B/2,SE);[/asy]
$\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$
2010 Contests, 2
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.
2012 All-Russian Olympiad, 4
Given is a pyramid $SA_1A_2A_3\ldots A_n$ whose base is convex polygon $A_1A_2A_3\ldots A_n$. For every $i=1,2,3,\ldots ,n$ there is a triangle $X_iA_iA_{i+1} $ congruent to triangle $SA_iA_{i+1}$ that lies on the same side from $A_iA_{i+1}$ as the base of that pyramid. (You can assume $a_1$ is the same as $a_{n+1}$.) Prove that these triangles together cover the entire base.
2013 Tuymaada Olympiad, 7
Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$.
Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$.
[i] A. Kupavsky [/i]
2008 AIME Problems, 15
A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$.
[asy]import math;
unitsize(5mm);
defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7));
pair O=(0,0);
pair A=(0,sqrt(17));
pair B=(sqrt(17),0);
pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75));
pair D=(xpart(C),8);
pair E=(8,ypart(C));
draw(O--(0,8));
draw(O--(8,0));
draw(O--C);
draw(A--C--B);
draw(D--C--E);
label("$\sqrt{17}$",(0,2),W);
label("$\sqrt{17}$",(2,0),S);
label("cut",midpoint(A--C),NNW);
label("cut",midpoint(B--C),ESE);
label("fold",midpoint(C--D),W);
label("fold",midpoint(C--E),S);
label("$30^\circ$",shift(-0.6,-0.6)*C,WSW);
label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]
2008 Harvard-MIT Mathematics Tournament, 1
How many different values can $ \angle ABC$ take, where $ A,B,C$ are distinct vertices of a cube?
1998 Brazil Team Selection Test, Problem 1
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
2009 Iran MO (3rd Round), 7
A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge.
Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces.
Time allowed for this problem was 1 hour.
1957 Moscow Mathematical Olympiad, 358
The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.
1973 IMO Shortlist, 6
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
Kvant 2019, M2580
We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic.
[i] Tibor Bakos and Géza Kós [/i]
1996 Hungary-Israel Binational, 2
$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.
2003 Romania National Olympiad, 4
In tetrahedron $ ABCD$, $ G_1,G_2$ and $ G_3$ are barycenters of the faces $ ACD,ABD$ and $ BCD$ respectively.
(a) Prove that the straight lines $ BG_1,CG_2$ and $ AG_3$ are concurrent.
(b) Knowing that $ AG_3\equal{}8,BG_1\equal{}12$ and $ CG_2\equal{}20$ compute the maximum possible value of the volume of $ ABCD$.
2020 Stanford Mathematics Tournament, 5
Find the smallest possible number of edges in a convex polyhedron that has an odd number of edges in total has an even number of edges on each face.
1957 Czech and Slovak Olympiad III A, 2
Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$.
(1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$.
(2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$
1962 IMO, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
2004 AMC 12/AHSME, 19
A truncated cone has horizontal bases with radii $ 18$ and $ 2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 4\sqrt5 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 6\sqrt3$
2013 Today's Calculation Of Integral, 891
Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space.
Let $V$ be the cone obtained by rotating the triangle around the $x$-axis.
Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.
1977 Czech and Slovak Olympiad III A, 1
There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.
1983 Putnam, B1
Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.
2007 AMC 12/AHSME, 16
Each face of a regular tetrahedron is painted either red, white or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 81$
2000 Harvard-MIT Mathematics Tournament, 3
Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?
2012 AMC 10, 17
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $
1977 Polish MO Finals, 1
Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.