Found problems: 85335
Let $AD$ be the inner angle bisector of the triangle $ABC$. The perpendicular on the side $BC$ at the point $D$ intersects the outer bisector of $\angle CAB$ at point $I$. The circle with center $I$ and radius $ID$ intersects the sides $AB$ and $AC$ at points $F$ and $E$ respectively. $A$-symmedian of $\Delta AFE$ intersects the circumcircle of $\Delta AFE$ again at point $X$. Prove that the circumcircles of $\Delta AFE$ and $\Delta BXC$ are tangent.
When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$?
${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2u \qquad\textbf{(C)}\ 3u \qquad\textbf{(D)}\ v }\qquad\textbf{(E)}\ 2v } $
Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.
Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.
Henry places some rooks and some kings in distinct cells of a $2\times 8$ grid such that no two rooks attack each other and no two kings attack each other. Find the maximum possible number of pieces on the board.
(Two rooks [i]attack[/i] each other if they are in the same row or column and no pieces are between them. Two kings attack each other if their cells share a vertex.)
Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.
$\definecolor{A}{RGB}{70,255,50}\color{A}\fbox{C6.}$ There are $n$ $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ and $n$ $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ in a club. Some of them are friends with each other. The $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ want to get into a [i]relationship[/i], so some subset of them wants to ask some $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ out for a trip. Because the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ are shy, for a nonempty set $B$ of $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$, they want to make sure that each of the girl they ask out is friend with one of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$. If the number of $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ they are able to ask out is smaller than the number of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$, then the nonempty set $B$ of those $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ is called a group of complete losers.
Show that for any $0 \le k < 2n$, there exists an arrangement of the [i]friendships[/i] among those $2n$ people so that there are exactly $k$ groups of complete losers.
[i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b].
[color=#3D9186]#1737[/color]
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers $1$ gallon/unit $^2$, and Robert starts at $(1, 0)$. Each second, he moves in a straight line from the point $(\cos(\theta),\sin(\theta))$ to the point $(\cos(\theta+a),\sin(\theta+a))$, where a changes after each movement. a starts out as $253^o$ and decreases by $2^o$ each step. If he takes $89$ steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as $\frac{\sqrt{a}- \sqrt{b}}{c} \cot 1^o$, where $a, b$ and $c$ are positive integers and no prime divisor of $c$ divides both $a$ and $b$ twice. Find $a + b + c$.
Let $\{a_n\}_{n \geq 1}$ be a bounded sequence satisfying
\[a_n < \displaystyle\sum_{k=a}^{2n+2006} \frac{a_k}{k+1} + \frac{1}{2n+2007} \quad \forall \quad n = 1, 2, 3, \ldots \]
Show that
\[a_n < \frac{1}{n} \quad \forall \quad n = 1, 2, 3, \ldots\]
Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$
Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer.
Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$
Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations
$x^2+y^2+25z^2=6xz+8yz$
$ 3x^2+2y^2+z^2=240$
Let $a,b,c,d$ be positive real numbers such that $abcd=4$ and
\[a^2+b^2+c^2+d^2=10.\]
Find the maximum possible value of $ab+bc+cd+da$.
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)-f(y)| \leq |x-y|$, for all $x,y \in \mathbb{R}$.
Prove that if for any real $x$, the sequence $x,f(x),f(f(x)),\ldots$ is an arithmetic progression, then there is $a \in \mathbb{R}$ such that $f(x)=x+a$, for all $x \in \mathbb R$.
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets: $\mathbb R^+ = A \cup B\cup C$ and $A \cap B = B \cap C = C \cap A = \emptyset$ whereas none of $A, B, C$ is empty.
[list=a][*] Show that one can choose $a \in A, b \in B$ and $c \in C$ such that $a,b, c$ are the sides of a triangle.
[*] Is it always possible to choose three numbers from three different sets $A,B,C$ such that these three numbers are the sides of a right-angled triangle?[/list]
Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$.$\omega$ intersects $\alpha$ at $A$ and $B$.Let $P$ any point on the circumference $\omega$.The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$.
Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$.
Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum.
[i]I. Juhasz[/i]
We say that the real numbers $a$ and $b$ have property $P$ if: $a^2+b \in Q$ and $b^2 + a \in Q$.Prove that:
a) The numbers $a= \frac{1+\sqrt2}{2}$ and $b= \frac{1-\sqrt2}{2}$ are irrational and have property $P$
b) If $a, b$ have property $P$ and $a+b \in Q -\{1\}$, then $a$ and $b$ are rational numbers
c) If $a, b$ have property $P$ and $\frac{a}{b} \in Q$, then $a$ and $b$ are rational numbers.
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satidfy :
$$2f(x+y+xy)= a f(x)+ bf(y)+f(xy)$$ for any $x,y \in\mathbb{R}$ όπου $a,b\in\mathbb{R}$ with $a^2-a\ne b^2-b$
Let $a_1$, $a_2$, $\ldots$, $a_n$ be a sequence of integers with values between 2 and 1995 such that:
(i) Any two of the $a_i$'s are relatively prime,
(ii) Each $a_i$ is either a prime or a product of primes.
Determine the smallest possible values of $n$ to make sure that the sequence will contain a prime number.
A set system $ (S,L)$ is called a Steiner triple system, if $ L\neq\emptyset$, any pair $ x,y\in S$, $ x\neq y$ of points lie on a unique line $ \ell\in L$, and every line $ \ell\in L$ contains exactly three points. Let $ (S,L)$ be a Steiner triple system, and let us denote by $ xy$ the thrid point on a line determined by the points $ x\neq y$. Let $ A$ be a group whose factor by its center $ C(A)$ is of prime power order. Let $ f,h: S\to A$ be maps, such that $ C(A)$ contains the range of $ f$, and the range of $ h$ generates $ A$.
Show, that if
\[ f(x) \equal{} h(x)h(y)h(x)h(xy)\]
holds for all pairs $ x\neq y$ of points, then $ A$ is commutative, and there exists an element $ k\in A$, such that $ f(x) \equal{} kh(x)$ for all $ x\in S$.
Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.
If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?