Found problems: 85335
Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent:
$ \text{(i)}\quad aba=a\wedge ba^2b=1 $
$ \text{(ii)}\quad ab=ba=1 $
$ \text{(iii)}\quad \exists !b\in A\quad aba=a $
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.
Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.
Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$.
Prove $|AP|=|AQ|$.
[i](Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)[/i]
Let be given $ \vartriangle ABC$ with area $ (\vartriangle ABC) = 60$ cm$^2$. Let $R,S $ lie in $BC$ such that $BR = RS = SC$ and $P,Q$ be midpoints of $AB$ and $AC$, respectively. Suppose that $PS$ intersects $QR$ at $T$. Evaluate area $(\vartriangle PQT)$.
5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by winnertakeover and Monkey_king1[/i]
Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where :
$$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$
Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]
Let $n$ integers on the real axis be colored. Determine for which positive integers $k$ there exists a family $K$ of closed intervals with the following properties:
i) The union of the intervals in $K$ contains all of the colored points;
ii) Any two distinct intervals in $K$ are disjoint;
iii) For each interval $I$ at $K$ we have ${{a}_{I}}=k.{{b}_{I}}$, where ${{a}_{I}}$ denotes the number of integers in $I$, and ${{b}_{I}}$ the number of colored integers in $I$.
Consider a set of $6$ fixed points in the plane, with no three collinear. Between some pairs of these points, we may draw one arrow from one point to the other. How many possible configurations of arrows are there such that if there is an arrow from point $A$ to point $B$ and an arrow from $B$ to $C,$ then there is an arrow from $A$ to $C?$ Your score will be $16-\frac{1}{800}|\textbf{Your Answer}-\textbf{Actual Answer}|$ rounded to the nearest integer or zero, whichever is higher.
For each positive integer $ n,$ let $ S(n)$ denote the sum of the digits of $ n.$ For how many values of $ n$ is $ n \plus{} S(n) \plus{} S(S(n)) \equal{} 2007?$
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
In an electronic game of questions and answers, for each correct answer the player adds $5$ points on the screen, for each incorrect answer $2$ points are subtracted and when the player does not answer, no score is added or subtracted. Each game has $30$ questions. Francisco played $5$ games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?
Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.
An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$?
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane.
When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.
$a$ and $b$ are given positive integers. Prove that there are infinitely many positive integers $n$ such that $n^b+1$ doesn't divide $a^n+1$.
For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.
Given an integer $n\geq 2$. Call a positive integer ${T}$ [i]Pingsheng Number[/i], if there exists pairwise different
non empty subsets $A_1,A_2,\cdots ,A_m$ $(m\geq 3)$ of set $S=\{1,2,\cdots ,n\},$ satisfying $T=\sum\limits_{i=1}^m|A_i|,$
and for $\forall p,q,r\in\{1,2,\cdots ,m\},p\neq q,q\neq r,r\neq p,$ we have $A_p\cap(A_q\triangle A_r)=\varnothing$ or $A_p\subseteq (A_q\triangle A_r).$
Find the max [i]Pingsheng Number[/i].
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column?
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
Let $\mathcal L$ be the set of all lines in the plane and let $f$ be a function that assigns to each line $\ell\in\mathcal L$ a point $f(\ell)$ on $\ell$. Suppose that for any point $X$, and for any three lines $\ell_1,\ell_2,\ell_3$ passing through $X$, the points $f(\ell_1),f(\ell_2),f(\ell_3)$, and $X$ lie on a circle.
Prove that there is a unique point $P$ such that $f(\ell)=P$ for any line $\ell$ passing through $P$.
[i]Australia[/i]
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$
I.Voronovich
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have?