Found problems: 85335
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.
[i]Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand[/i]
Find all integers \( n > 1 \) such that all prime divisors of \( n^6 - 1 \) divide \( (n^2 - 1)(n^3 - 1) \).
How many real roots do $x^5 +3x^4 -4x^3 -8x^2 +6x-1$ and $x^5-3x^4 -2x^3 -3x^2 -6x+1$ share?
Given a real number $t\ge3$, suppose a polynomial $f\in\mathbb R[x]$ satisfies
$$\left|f(k)-t^k\right|<1,\enspace\forall k=0,1,\ldots,n.$$Prove that $\deg f\ge n$.
Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.
Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers.
[i]Proposed by ABCDE and Tristan Shin
Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$, $BC$, or $CA$. Distinct points $D$, $E$, and $F$ lie on lines $BC$, $AC$, and $AB$, respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$. Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$.
[i]Andrew Gu[/i]
Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.
Given real numbers $x, y, z, w$ such that $(x + y + 2z)(x + z + 3w) = 1$, what is the minimum possible value of $x^2 + y^2 + z^2 + w^2$?
The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.
Let $ABCD$ be a cyclic quadrilateral with all distinct sides that has an inscribed circle. The incircle of $ABCD$ has center $I$ and is tangent to $AB$, $BC$, $CD$, and $DA$ at points $W$, $X$, $Y$, and $Z$, respectively. Let $K$ be the intersection of the lines $WX$ and $YZ$. Prove that $KI$ is tangent to the circumcircle of triangle $AIC$.
$f : \mathbb{Z} \rightarrow \mathbb{Z}$ satisfies $m+f(m+f(n+f(m))) = n + f(m)$ for every integers $m,n$.
If $f(6) = 6$, then $f(2012) = ?$
$ \textbf{(A)}\ -2010 \qquad \textbf{(B)}\ -2000 \qquad \textbf{(C)}\ 2000 \qquad \textbf{(D)}\ 2010 \qquad \textbf{(E)}\ 2012$
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.
For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.
(B.Frenkin) Does a convex quadrilateral without parallel sidelines exist such that it can be divided into four congruent triangles?
Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 91\qquad
\textbf{(C)}\ 133\qquad
\textbf{(D)}\ 195\qquad
\textbf{(E)}\ 211$
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$
\[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\]
Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.
In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?
Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality:
\[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
$$
f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3
$$
for all integers $a, b, c$.
For each positive number $x$, let \[f(x)=\displaystyle\frac{\left(x+\dfrac{1}{x}\right)^6-\left(x^6+\dfrac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}.\] The minimum value of $f(x)$ is
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }6$
Determine all the quadruples of real numbers that satisfy the following:
[i]The product of any three of these numbers plus the fourth is constant.[/i]
Find all positive integers $n$ such that
$$n(2^n-1)$$
is a perfect square
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$.
[i]Proposed by Ankan Bhattacharya, USA[/i]