Found problems: 2265
2012-2013 SDML (Middle School), 12
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
1983 Brazil National Olympiad, 2
An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.
2005 Taiwan TST Round 1, 2
Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.
1994 BMO TST – Romania, 4:
Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$
1998 Mediterranean Mathematics Olympiad, 2
Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.
1960 AMC 12/AHSME, 24
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:
$ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$
$\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$
$\textbf{(C)}\ \text{An irrational number} \qquad$
$\textbf{(D)}\ \text{A perfect square}\qquad$
$\textbf{(E)}\ \text{A perfect cube} $
1991 AMC 12/AHSME, 14
If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be
$ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $
1971 IMO Shortlist, 7
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.
[b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;
[b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
1974 IMO Longlists, 6
Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
KoMaL A Problems 2022/2023, A.837
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that
\[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \]
[i]Submitted by Viktor Vígh, Szeged[/i]
1996 Romania National Olympiad, 2
Let $ABCD$ a tetrahedron and $M$ a variable point on the face $BCD$. The line perpendicular to $(BCD)$ in $M$ . intersects the planes$ (ABC)$, $(ACD)$, and $(ADB)$ in $M_1$, $M_2$, and $M_3$. Show that the sum $MM_1 + MM_2 + MM_3$ is constant if and only if the perpendicular dropped from $A$ to $(BCD)$ passes through the centroid of triangle $BCD$.
2002 IMC, 10
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.
1975 Vietnam National Olympiad, 3
Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.
2002 National Olympiad First Round, 24
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ \text{Infinitely many}
\qquad\textbf{e)}\ \text{None of above}
$
1937 Moscow Mathematical Olympiad, 036
* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?
2018 AMC 8, 24
In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$
[asy]
size(6cm);
pair A,B,C,D,EE,F,G,H,I,J;
C = (0,0);
B = (-1,1);
D = (2,0.5);
A = B+D;
G = (0,2);
F = B+G;
H = G+D;
EE = G+B+D;
I = (D+H)/2; J = (B+F)/2;
filldraw(C--I--EE--J--cycle,lightgray,black);
draw(C--D--H--EE--F--B--cycle);
draw(G--F--G--C--G--H);
draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J);
label("$A$",A,E);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$E$",EE,N);
label("$F$",F,W);
label("$G$",G,N);
label("$H$",H,E);
label("$I$",I,E);
label("$J$",J,W);
[/asy]
$\textbf{(A) } \frac{5}{4} \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{25}{16} \qquad \textbf{(E) } \frac{9}{4}$
1980 Putnam, B2
Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six
simultaneous conditions:
$$ x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.$$
a) Determine the number $V$ of vertices of $S.$
b) Determine the number $E$ of edges of $S.$
c) Sketch in the $bc$-plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$
2000 Denmark MO - Mohr Contest, 3
A [i]Georg Mohr[/i] cube is a cube with six faces printed respectively $G, E, O, R, M$ and $H$. Peter has nine identical Georg Mohr dice. Is it possible to stack them on top of each other for a tower there on each of the four pages in some order show the letters $G\,\, E \,\, O \,\, R \,\, G \,\, M \,\, O \,\, H \,\, R$?
2013 AMC 10, 24
A positive integer $n$ is [i]nice[/i] if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numers in the set $\{2010, 2011, 2012,\ldots,2019\}$ are nice?
${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D}}\ 4\qquad\textbf{(E)}\ 5 $
2004 National High School Mathematics League, 6
Shaft section of a circular cone with vertex $P$ is an isosceles right triangle. $A$ is a point on the circle of the bottom surface, while $B$ is a point inside the circle, $O$ is the center of the circle. If $AB\perp OB$ at $B$, $OH\perp PB$ at $H$, $PA=4$, $C$ is the midpoint of $PA$, then when the volume of triangular pyramid $O-HPC$ takes its maximum value, the length of $OB$ is
$\text{(A)}\frac{\sqrt5}{3}\qquad\text{(B)}\frac{2\sqrt5}{3}\qquad\text{(C)}\frac{\sqrt6}{3}\qquad\text{(D)}\frac{2\sqrt6}{3}\qquad$
2012 Poland - Second Round, 2
Prove that for tetrahedron $ABCD$; vertex $D$, center of insphere and centroid of $ABCD$ are collinear iff areas of triangles $ABD,BCD,CAD$ are equal.
1979 IMO, 1
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
IV Soros Olympiad 1997 - 98 (Russia), 11.4
Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).
2016 PUMaC Team, 13
Ayase randomly picks a number $x \in (0, 1]$ with uniform probability. He then draws the six points $(0, 0, 0),(x, 0, 0),(2x, 3x, 0),(5, 5, 2),(7, 3, 0),(9, 1, 4)$. If the expected value of the volume of the convex polyhedron formed by these six points can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$
1973 AMC 12/AHSME, 32
The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 9/2 \qquad
\textbf{(C)}\ 27/2 \qquad
\textbf{(D)}\ \frac{9\sqrt3}{2} \qquad
\textbf{(E)}\ \text{none of these}$