Found problems: 2265
Maryland University HSMC part II, 2015
[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins.
(a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$.
(b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$.
[b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation
$$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012}
= \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$
[b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets.
[b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance.
(a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$.
(b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$.
[b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 All-Russian Olympiad Regional Round, 11.8
3 segments $AA_1$, $BB_1$, $CC_1$ in space share a common midpoint $M$. Turns out, the sphere circumscribed about the tetrahedron $MA_1B_1C_1$ is tangent to plane $ABC$ at point $D$. Point $O$ is the circumcenter of triangle $ABC$. Prove that $MO = MD$.
2021 Science ON all problems, 3
$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that
$$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$
[i] (Adapted from folklore)[/i]
2000 Harvard-MIT Mathematics Tournament, 5
Find all $3$-digit numbers which are the sums of the cubes of their digits.
V Soros Olympiad 1998 - 99 (Russia), 11.10
The plane angles at vertex $D$ of the pyramid $ABCD$ are equal to $\alpha$,$\beta$ and $\gamma$ ($\angle CDB = a$). An arbitrary point $M$ is taken on edge $CB$. A ball is inscribed in each of the pyramids $ABDM$ and $ACDM$. Let us draw through $D$ a plane distinct from $BCD$, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment $AM$ at point $P$. What is $\angle ADP$ equal to?
2003 National Olympiad First Round, 10
Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 17
\qquad\textbf{(D)}\ 22
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1963 Miklós Schweitzer, 1
Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]
1991 French Mathematical Olympiad, Problem 3
Let $S$ be a fixed point on a sphere $\Sigma$ with center $\Omega$. Consider all tetrahedra $SABC$ inscribed in $\Sigma$ such that $SA,SB,SC$ are pairwise orthogonal.
(a) Prove that all the planes $ABC$ pass through a single point.
(b) In one such tetrahedron, $H$ and $O$ are the orthogonal projections of $S$ and $\Omega$ onto the plane $ABC$, respectively. Let $R$ denote the circumradius of $\triangle ABC$. Prove that $R^2=OH^2+2SH^2$.
2002 Flanders Math Olympiad, 1
Is it possible to number the $8$ vertices of a cube from $1$ to $8$ in such a way that the value of the sum on every edge is different?
2014 Contests, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
1940 Putnam, B4
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
$$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$
is the sphere
$$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
1988 IMO Longlists, 32
$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.
1954 Polish MO Finals, 5
Prove that if in a tetrahedron $ ABCD $ opposite edges are equal, i.e. $ AB = CD $, $ AC = BD $, $ AD = BC $, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are the axes of symmetry of the tetrahedron.
2006 China Team Selection Test, 1
Let $K$ and $M$ be points on the side $AB$ of a triangle $\triangle{ABC}$, and let $L$ and $N$ be points on the side $AC$. The point $K$ is between $M$ and $B$, and the point $L$ is between $N$ and $C$. If $\frac{BK}{KM}=\frac{CL}{LN}$, then prove that the orthocentres of the triangles $\triangle{ABC}$, $\triangle{AKL}$ and $\triangle{AMN}$ lie on one line.
1998 Putnam, 1
A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?
1991 IMTS, 5
Show that it is impossible to dissect an arbitary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.
1995 May Olympiad, 4
Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?
KoMaL A Problems 2017/2018, A. 724
A sphere $S$ lies within tetrahedron $ABCD$, touching faces $ABD, ACD$, and $BCD$, but having no point in common with plane $ABC$. Let $E$ be the point in the interior of the tetrahedron for which $S$ touches planes $ABE$, $ACE$, and $BCE$ as well. Suppose the line $DE$ meets face $ABC$ at $F$, and let $L$ be the point of $S$ nearest to plane $ABC$. Show that segment $FL$ passes through the centre of the inscribed sphere of tetrahedron $ABCE$.
KöMaL A.723. (April 2018), G. Kós
2011 CentroAmerican, 1
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
2001 Pan African, 2
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
2008 Harvard-MIT Mathematics Tournament, 28
Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.
1968 Polish MO Finals, 3
In a tetrahedron $ ABCD $ the edges $AD$, $ BD $, $ CD $ are equal. $ ABC $ Non-collinear points are chosen in the plane. $ A_1$, $B_1$, $C_1 $ The lines $DA_1$, $DB_1$, $DC_1 $ intersect the surface of the sphere circumscribed about the tetrahedron at points $ A_2$, $B_2$, $C_2 $, different from the point $ D $. Prove that the points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie on the surface of a certain sphere.
2005 Brazil National Olympiad, 3
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$.
2015 BMT Spring, 9
Find the side length of the largest square that can be inscribed in the unit cube.
2020 Tournament Of Towns, 5
A triangle is given on a sphere of radius $1$, the sides of which are arcs of three different circles of radius $1$ centered in the center of a sphere having less than $\pi$ in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere.
A. Zaslavsky