Found problems: 85335
Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.
In tetrahedron $ABCD$, $AB=1,CD=3$, the distance between $AB$ and $CD$ is $2$, the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, then the volume of tetrahedron $ABCD$ is
$\text{(A)}\frac{\sqrt3}{2}\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}\frac{\sqrt3}{3}$
$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.
Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)
Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.
Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that:
$i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$.
$ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$.
a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist.
b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.
Consider a computer network consisting of servers and bi-directional communication channels among them. Unfortunately, not all channels operate. Each direction of each channel fails with probability $p$ and operates otherwise. (All of these stochastic events are mutually independent, and $0 \le p \le 1$.) There is a root serve, denoted by $r$. We call the network [i]operational[/i], if all serves can reach $r$ using only operating channels. Note that we do not require $r$ to be able to reach any servers.
Show that the probability of the network to be operational does not depend on the choice of $r$. (In other words, for any two distinct root servers $r_1$ and $r_2$, the operational probability is the same.)
Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that $\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0$. Note: for a prime $p$ and a positive integer $m$, $v_p(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $v_3(18) = 2$.
In how many ways can we construct a square with dimensions $4\times 4$ using $4$ white, $4$ green , $4$ red and 4 $blue$ squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .
In a chess tournament each player played one match with every other player. It is known that all players have different scores. The player who is on the last place got $k$ points. What is the smallest number of wins that the first placed player got? (For the win $1$ point is given, for loss $0$ and for a draw both players get $0,5$ points.)
Kelvin the Frog and Alex the Kat are playing a game on an initially empty blackboard. Kelvin begins by writing a digit. Then, the players alternate inserting a digit anywhere into the number currently on the blackboard, including possibly a leading zero (e.g. $12$ can become $123$, $142$, $512$, $012$, etc.). Alex wins if the blackboard shows a perfect square at any time, and Kelvin's goal is prevent Alex from winning. Does Alex have a winning strategy?
Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear.
[i](Proposed by Tan Rui Xuen)[/i]
In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?
(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3.
(b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.
Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.
The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.
Let $ a $ and $ b $ be integer numbers. Let $ a = a^{2}+b^{2}-8b-2ab+16$. Prove that $ a $ is a perfect square.
A [i]word[/i] is defined as any finite string of letters. A word is a [i]palindrome[/i] if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.
Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.
We have squares $ABCD$ and $EFGH$. Square $ABCD$ has points with coordinates $A=(1,1,-1)$, $B=(1,-1,-1)$, $C=(-1,-1,-1)$ and $D=(-1,1,-1)$. Square $EFGH$ has points with coordinates $E=(\sqrt{2},0,1)$, $F=(0,-\sqrt{2},1)$, $G=(-\sqrt{2},0,1)$, and $H=(0,\sqrt{2},1)$. Consider the solid formed by joining point $A$ to $H$ and $E$, point $B$ to $E$ and $F$, point $C$ to $F$ and $G$, and point $D$ to $G$ and $H$. Compute the volume of this solid.
Given a set of $n$ positive numbers. For each its nonempty subset consider the sum of all the subset's numbers. Prove that you can divide those sums onto $n$ groups in such a way, that the least sum in every group is not less than a half of the greatest sum in the same group.
Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.
Triangle $ABC$ has $BC<AC$ and circumradius $8$. Let $O$ be the circumcenter of $\triangle ABC$, $M$ be the midpoint of minor arc $AB$, and $C'$ be the reflection of $C$ across $OM$. If $AB$ bisects $\angle OAM$, and $\angle COC' = 120^\circ$, find the square of the area of the convex pentagon $CC'AMB$.
[i]Team #4[/i]
A plane $p$ passes through a vertex of a cube so that the three edges at the vertex make equal angles with $p$. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto $p$. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane $p$.
Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.