Found problems: 85335
Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.
Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.
$ABCD$ is quadrilateral inscribed in a circle $\Gamma$ .Lines $AB$ and $CD$ intersect at $E$ and lines$AD$ and $BC$ intersect at $F$.
Prove that the circle with diameter $EF$ and circle $\Gamma$ are orthogonal.
If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, $\cdots$, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 13
\qquad\textbf{(C)}\ 14
\qquad\textbf{(D)}\ 15
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Determine if there exists an infinite sequence $a_0, a_1, a_2, a_3,...$ of nonegative integers that satisfies the following conditions:
(i) All nonegative integers appear in the sequence exactly once.
(ii) The succession
$b_n=a_{n}+n,$, $n\geq0$,
is formed by all prime numbers and each one appears exactly once.
There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules.
(i) Each players move the stone to a neighboring intersection along a segment, by turns.
(ii) A segment, which is already passed by the stone, cannot be used more.
(iii) One who cannot move the stone anymore loses.
Prove that there is a winning strategy for the former player.
The USAMTS tug-of-war team needs to pick a representative to send to the national tug-of-war convention. They don't care who they send, as long as they don't send the weakest person on the team. Their team consists of $20$ people, who each pull with a different constant strength. They want to design a tournament, with each round planned ahead of time, which at the end will allow them to pick a valid representative. Each round of the tournament is a $10$-on-$10$ tug-of-war match. A round may end in one side winning, or in a tie if the strengths of each side are matched. Show that they can choose a representative using a tournament with $10$ rounds.
If $ a$ is a nonzero integer and $ b$ is a positive number such that $ ab^{2} \equal{} \log_{10}b,$ what is the median of the set $ \{0,1,a,b,1/b\}$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ a \qquad \textbf{(D)}\ b \qquad \textbf{(E)}\ \frac {1}{b}$
Do either $(1)$ or $(2)$
$(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$.
$(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$
Two truth tellers and two liars are positioned in a line, where every person is distinguishable. How many ways are there to position these four people such that everyone claims that all people directly adjacent to them are liars?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }12\qquad\textbf{(E) }16$
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
[i]Proposed by Abbas Mehrabian, Iran[/i]
Determine all pairs $(x, y)$ of positive integers satisfying
$x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.
For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is:
$\textbf{(A)}\ R+R'D\qquad
\textbf{(B)}\ R'+RD\qquad
\textbf{(C)}\ RR'\qquad
\textbf{(D)}\ R\qquad
\textbf{(E)}\ R'$
Find all pairs of positive integers $(m,n)$ such that
\[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]
How many ways are there to arrange three indistinguishable rooks on a $ 6 \times 6$ board such that no two rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)
A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.
Let $\triangle ABC$ be an isosceles.$(AB=AC)$.Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$,$E\in DC$ and $2\angle DAE = \angle BAC$.Prove that we can construct a triangle $XYZ$ such that $XY=BD$,$YZ=DE$ and $ZX=EC$.Find $\angle BAC + \angle YXZ$.
Let $a_1,a_2,\ldots$ be a sequence defined by $a_1 = 1$ and for $n\ge1$, $a_{n+1} = \sqrt{a_n^2 -2a_n + 3} + 1$. Find $a_{513}$.
[i]Ray Li.[/i]
If
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$
where $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$
In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.
A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.
A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy
$$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$$
for every $n\geq 1$. Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$.
Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality:
\[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.
The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.