Found problems: 85335
Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$?
(N.Beluhov)
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
1. Let $f(x)=x^2+bx+c$, M = {x | |f(x)|<1}. Prove $|M|\leq 2\sqrt{2}$ (|...| = length of interval(s))
A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.
For a natural number $ n\ge 2, $ prove that the $ \text{n-ary} $ direct product of the group of order $ 2 $ is abelian and isomorphic with the group of the power set of a set under symmetric difference.
We say that a set $M{}$ containing $4$ elements is "evenly connected" if for each element $x\in M$, at least one of the numbers $x-2$ or $x+2$ belongs to the set $M.$ Let $S_n$ be the number of "evenly connected" subsets of $\{1,2,3\ldots,n\}$. Find the smallest $n{}$ such that $S_n \geq 2011.$
Let $S = \{1,2,3,\dots,2014\}$. What is the largest subset of $S$ that contains no two elements with a difference of $4$ or $7$?
Consider the function $f(x, y, z) = (x-y +z,y -z +x, z-x+y)$ and denote by $f^{(n)}(x, y,z)$ the function $f$ applied $n$ times to the tuple $(x,y,z)$. Let $r_1$, $r_2$, $r_3$ be the three roots of the equation $x^3- 4x^2 + 12 = 0$ and let $g(x) = x^3 + a_2x^2 + a_1x + a_0$ be the cubic polynomial with the tuple $f^{(3)}(r_1, r_2, r_3)$ as roots. Find the value of $a_1$.
For a sale, a store owner reduces the price of a $10$ dollar scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now
$ \text{(A)}\ 2.00\text{ dollars}\qquad\text{(B)}\ 3.75\text{ dollars}\qquad\text{(C)}\ 4.00\text{ dollars}\qquad\text{(D)}\ 4.90\text{ dollars}\qquad\text{(E)}\ 6.40\text{ dollars} $
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$
[i]V. Gordon[/i]
If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that
$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers.
i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$
ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples.
[i]Dan Schwarz[/i]
Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there is the same number of both types of cars. He can try to fit together two cars in one try. What is the least number of tries needed?
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
We have a bipartite graph $G$ (with parts $X$ and $Y$). We orient each edge arbitrarily. [i]Hessam[/i] chooses a vertex at each turn and reverse the orientation of all edges that $v$ is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex $v$ in part $X$, $\deg^{+}(v)\geq \deg^{-}(v)$ and for each vertex in part $Y$, $\deg^{+}v\leq \deg^{-}v$
Given equation ${a^5} - {a^3} + a = 2$, with real $a$ . Prove that $3 < {a^6} < 4$.
Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$. Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$. Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$. If the radius of $\Gamma$ is $17$, the radius of $\omega$ is $7$, and $\frac{AQ}{BQ}=3$, find the circumradius of triangle $AQR$.
Let $n \ge 2$ be an integer and let $a_1, a_2, \cdots a_n$ be fixed positive integers (not necessarily all distinct) in such a way that $\gcd(a_1, a_2 \cdots a_n)=1$. In a board the numbers $a_1, a_2 \cdots a_n$ are all written along with a positive integer $x$. A move consists of choosing two numbers $a>b$ from the $n+1$ numbers in the board and replace them with $a-b,2b$. Find all possible values of $x$, with respect of the values of $a_1, a_2 \cdots a_n$, for which it is possible to achieve a finite sequence of moves (possibly none) such that eventually all numbers written in the board are equal.
At a table tennis competition, every pair of players played each other exactly once. Every boy beat thrice as many boys as girls, and every girl was beaten by twice as many girls as boys. How many competitors were there, if we know that there were $10$ more boys than girls?
There are no draws in table tennis, every match was won by one of the two players.
(i) Find a formula for $S_n = -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^n n^2 \times (n + 1)$ in terms of the positive integer $n$. Justify your answer.
(As an example, one has $1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$)
(ii) Using your formula in (i), find the value of
$ -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^{100} 100^2 \times (100 + 1)$
Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?
$p,q$ are nonnegative integers.Given two conditions:
A: $p^3-q^3$ is an even number.
B: $p+q$ is an even number.
Then, which one of the followings are true?
$(\text{A})$A is sufficient but unnecessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient and necessary condition of B.
$(\text{D})$A is insufficient and unnecessary condition of B.
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.