Found problems: 85335
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer.
Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$), where $k$ is an integer with $k \ge K$.
There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.
Let be a set $ A\in (0,\infty )\setminus\{ 1\} $ and two operations $ *,\circ :A^2\longrightarrow A $ defined as
$$ x*y=x^{2\log_3 y} ,\quad x\circ y= x^{3\log_2y} , $$
and chosen such that $ (A,*) , (A,\circ ) $ are groups. Prove that these groups are isomorphic.
[i]Gabriel Iorgulescu[/i]
Prove that for any two linear subspaces $V, W \subset R^n$ the same
dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$
A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.
The evil Dr. Doom seeks to destroy his enemy, the Intergalactic Federation, and has devised a plan to despin the Federation's space station. The hoop-shaped space station of mass $M$ and radius $R$ rotates once every $T$ hours to maintain artificial gravity equal to that on IPhOO. Dr. Doom plans to mount two thruster rockets, one rocket on opposite sides of the space station, to stop its rotation. Dr. Doom must accomplish his crime within a time $t$ to avoid getting caught. How much force should each rocket deliver in order to despin the Federation's space station in $t$? Express your answer in terms of $M$, $R$, $T$, $t$, and/or constants, as necessary.
[i]Problem proposed by Kimberly Geddes[/i]
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
Integers between $1$ and $7$ are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times.
A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set $\{1, 2, 3, 4, 5, 6 , 7\}$
For example, moves allowed are:
• delete a $4$ and a $5$, and write in their place the numbers $1, 2, 3, 6$ and $7$;
• deleting a $1$, a $2$, a $3$, a $4$, a $5$, a $6$ and a $7$ and write nothing in their place.
Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.
Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits
$a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$.
Let $z\neq0$ be a complex number satisfying $z^2=z+i|z|$. ($|z|$ denotes the length between the origin and $z$ in the complex plane.) Find $z\cdot\overline{z}$, where $\overline{z}=a-bi$ is the complex conjugate of $z=a+bi$.
Let $x=\frac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}$. The value of $$H=(1+x^5-x^7)^{{2012}^{3^{11}}}$$
is
(A) $1$ (B) $11$ (C) $21$ (D) $101$ (E) None of the above
Which number is larger: $A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}$, or $B = \log_{2023} 91125$?
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
A special kind of chess knight is in the origin of an infinite grid. It can make one of twelve different moves: it can move directly up, down, left, or right one unit square, or it can move $1$ units in one direction and $3$ units in an orthogonal direction. How many different squares can it be on after $2$ moves?
The sequence $S_0,S_1,S_2,\ldots$ is defined by[list][*]$S_n=1$ for $0\le n\le 2011$, and
[*]$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.[/list]Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$
How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side
lengths and perimeter less than 20?
Let $ABC$ be a scalene triangle with circle $\Gamma$. Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$. Let $U, V, W, Z$ be the intersections, distinct from $A$, of the $AP, AQ, AR$ and $AS$ with $\Gamma$, respectively. Let $X = UQ \cap SW$, $Y = PV \cap ZR$, $T = UR \cap VS$ and $K = PW \cap ZQ$. Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$. Show that $M, N, A$ are collinear.
Given a fixed integer $k>0,r=k+0.5$,define
$f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$
where $[x]$ denotes the smallest integer not less than $x$.
prove that there exists integer $m$ such that $f^m(r)$ is an integer.
Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$
The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.
Consider horizontal and vertical segments in the plane that may intersect each other. Let $n$ denote their total number. Suppose that we have $m$ curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that $m=O(n)$.
Let $m$ be a positive integer and let the lines $13x+11y=700$ and $y=mx-1$ intersect in a point whose coordinates are integers. Then $m$ is:
$\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ \text{one of the integers}~ 4,5,6,7~\text{and one other positive integer}$