Found problems: 85335
Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is \[ \frac{2ab\cos \frac{C}{2}}{a+b}. \]
A finite set $S{}$ of $n{}$ points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed $n{}$-link polylines with vertices at these points will be denoted by $f(S).$ Prove that
[list=a]
[*]$f(S)>0$ for all sets $S{};$
[*]$f(S)=1$ if and only if all the points of $S{}$ lie on the convex hull of $S{};$
[*]if $f(S)>1$ then $f(S)\geqslant n-1$, with equality if and only if one point of $S$ lies inside the convex hull;
[*]if exactly two points of $S{}$ lie inside the convex hull, then\[f(S)\geqslant\frac{(n-2)(n-3)}{2}.\]
[/list]Let $n\geqslant 3.$ Denote by $F(n)$ the largest possible value of the function $f(S)$ over all admissible sets $S{}$ of $n{}$ points. Prove that \[F(n)\geqslant3\cdot 2^{(n-8)/3}.\][i]Proposed by E. Bakaev and D. Magzhanov[/i]
Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)
[i](Birgit Vera Schmidt)[/i]
[hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person.
In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang.
Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]
Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that:
$$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$
The sum of the areas of all triangles whose vertices are also vertices of a $1\times 1 \times 1$ cube is $m+\sqrt{n}+\sqrt{p}$, where $m$, $n$, and $p$ are integers. Find $m+n+p$.
Let $ P$ be a point on the diagonal $ BD$ of a rectangle $ ABCD$, $ F$ be the projection of $ P$ on $ BC$, and $ H \not\equal{} B$ be the point on $ BC$ such that $ BF\equal{}FH$. If lines $ PC$ and $ AH$ intersect at $ Q$, prove that the areas of triangles $ APQ$ and $ CHQ$ are equal.
Consider a completely inelastic collision between two lumps of space goo. Lump 1 has mass $m$ and originally moves directly north with a speed $v_\text{0}$. Lump 2 has mass $3m$ and originally moves directly east with speed $v_\text{0}/2$. What is the final speed of the masses after the collision? Ignore gravity, and assume the two lumps stick together after the collision.
(A) $7/16 \ v_\text{0}$
(B) $\sqrt{5}/8 \ v_\text{0}$
(C) $\sqrt{13}/8 \ v_\text{0}$
(D) $5/8 \ v_\text{0}$
(E) $\sqrt{13/8} \ v_\text{0}$
Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.
Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.
The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?
66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.
In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that
[b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i];
[b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i].
Find the least possible number of [i]central[/i] vertices in $G$.
For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$
satisfying the two following properties:
[b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and
[b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$
[i]Proposed by Serbia[/i]
$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.
Mr. A owns a house worth $ \$10000$. He sells it to Mr. B at $ 10 \%$ profit. Mr. B sells the house back to Mr. A at a $ 10 \%$ loss. Then:
$ \textbf{(A)}\ \text{Mr. A comes out even} \qquad
\textbf{(B)}\ \text{Mr. A makes }\$100 \qquad
\textbf{(C)}\ \text{Mr. A makes }\$1000 \\
\textbf{(D)}\ \text{Mr. B loses }\$100 \qquad
\textbf{(E)}\ \text{none of the above is correct}$
Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]
If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$
Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of $\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}$
Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$
Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.
The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.
In right triangle $ ABC$, $ BC \equal{} 5$, $ AC \equal{} 12$, and $ AM \equal{} x$; $ \overline{MN} \perp \overline{AC}$, $ \overline{NP} \perp \overline{BC}$; $ N$ is on $ AB$. If $ y \equal{} MN \plus{} NP$, one-half the perimeter of rectangle $ MCPN$, then:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);
pair A = origin;
pair M = (1,0);
pair C = (2,0);
pair P = (2,0.5);
pair B = (2,1);
pair Q = (1,0.5);
draw(A--B--C--cycle);
draw(M--Q--P);
label("$A$",A,SW);
label("$M$",M,S);
label("$C$",C,SE);
label("$P$",P,E);
label("$B$",B,NE);
label("$N$",Q,NW);[/asy]$ \textbf{(A)}\ y \equal{} \frac {1}{2}(5 \plus{} 12) \qquad \textbf{(B)}\ y \equal{} \frac {5x}{12} \plus{} \frac {12}{5}\qquad \textbf{(C)}\ y \equal{} \frac {144 \minus{} 7x}{12}\qquad$
$ \textbf{(D)}\ y \equal{} 12\qquad \qquad\quad\,\, \textbf{(E)}\ y \equal{} \frac {5x}{12} \plus{} 6$