Found problems: 85335
Given an acute triangle $ABC$ with $AB<AC$, let $D$ be the foot of altitude from $A$ to $BC$ and let $M\neq D$ be a point on segment $BC$.$\,J$ and $K$ lie on $AC$ and $AB$ respectively such that $K,J,M$ lies on a common line perpendicular to $BC$. Let the circumcircles of $\triangle ABJ$ and $\triangle ACK$ intersect at $O$. Prove that $J,O,M$ are collinear if and only if $M$ is the midpoint of $BC$.
[i]Proposed by Wong Jer Ren[/i]
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$.
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[i](Vlad Robu)[/i]
Determine all integers $ n > 1$ such that
\[ \frac {2^n \plus{} 1}{n^2}
\]
is an integer.
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are
$\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$
A rhombus with sidelength $1$ has an inscribed circle with radius $\frac{1}{3}.$ If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive $a,b,$ evaluate $a+b.$
[i]Proposed by Alex Li[/i]
Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.
Let $\{a_i\}_{i=1}^{\infty}$ be a sequence of positive integers such that $a_1<a_2<a_3\cdots$ and all of primes are members of this sequence. Prove that for every $n<m$
\[\dfrac{1}{a_n} + \dfrac{1}{a_{n+1}} + \cdots + \dfrac{1}{a_m} \not \in \mathbb N\]
We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$
Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.
[i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]
Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at points $M$ and $N$. The radii of $\omega_1$ and $\omega_2$ are $12$ and $15$, respectively, and $O_1O_2 = 18$. A point $X$ is chosen on segment $MN$. Line $O_1X$ intersects $\omega_2$ at points $A$ and $C$, where $A$ is inside $\omega_1$. Similarly, line $O_2X$ intersects $\omega_1$ at points $B$ and $D$, where $B$ is inside $\omega_2$. The perpendicular bisectors of segments $AB$ and $CD$ intersect at point $P$. Given that $PO_1 = 30$, find $PO_2^2$.
[i]Proposed by Andrew Zhao[/i]
In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?
$ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $
Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.
Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$.
[i]Proposed by Evan Chang[/i]
Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
What is the perimeter of trapezoid $ ABCD$?
[asy]defaultpen(linewidth(0.8));size(3inch, 1.5inch);
pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0);
draw(a--b--c--d--cycle);
draw(b--e);
draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e);
label("30", (9,12), W);
label("50", (43,24), N);
label("25", (71.5, 12), E);
label("24", (18, 12), E);
label("$A$", a, SW);
label("$B$", b, N);
label("$C$", c, N);
label("$D$", d, SE);
label("$E$", e, S);[/asy]
$ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 188\qquad\textbf{(C)}\ 196\qquad\textbf{(D)}\ 200\qquad\textbf{(E)}\ 204 $
Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.
$99$ cards each have a label chosen from $1,2,\dots,99$, such that no (non-empty) subset of the cards has labels with total divisible by $100$. Show that the labels must all be equal.
Let $M$ be a set with $n$ elements. How many pairs $(A, B)$ of subsets of $M$ are there such that $A$ is a subset of $B?$
Let the line
$ z\equal{}x, \, y\equal{}0$
rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed.
(a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped).
(b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.
Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that:
$a)$ $f$ is continuous;
$b)$ $f$ is strictly monotone.