Found problems: 85335
Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation
\[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\]
for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.
Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
[i]Proposed by Sammy Charney[/i]
Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$
Let $a;b;c;d>0$ such that $abcd=1$. Prove that $a^2+b^2+c^2+d^2+a(b+c)+b(c+d)+c(d+a)\ge 10$
An Olympiad has 99 tasks. Several participants of the Olympiad are standing in a circle. They all solved different sets of tasks. Any two participants standing side by side do not have a common solved problem, but have a common unsolved one. Prove that the number of participants in the circle does not exceed \[2^{99}-\binom{99}{50}.\]
Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.
Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$.
[i]Proposed by Md. Ashraful Islam Fahim[/i]
Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.
The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner.
Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $
In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.
[i]Proposed by Holden Mui[/i]
Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre.
[i]Ion Pătrașcu, Ion Cotoi[/i]
The sum of $n > 2$ nonzero real numbers (not necessarily distinct) equals zero. For each of the $2^n - 1$ ways to choose one or more of these numbers, their sums are written in non-increasing order in a row. The first number in the row is $S$. Find the smallest possible value of the second number.
As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along. Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others. After a while, Tony comes to a problem he likes from an old AMC 10 exam:
\begin{align*}&\text{Four distinct circles are drawn in a plane. What is the maximum}\\&\quad\,\,\text{number of points where at least two of the circles intersect?}\end{align*}
Tony realizes that he can draw the four circles such that each pair of circles intersects in two points. After careful doodling, Tony finds the correct answer, and is proud that he can solve a problem from late on an AMC 10 exam.
"Mom, why didn't we all get Tony's brain?" Wendy inquires before turning he head back into her favorite Harry Potter volume (the fifth year).
Joshua leans over to Tony's seat to see his brother's work. Joshua knows that Tony has not yet discovered all the underlying principles behind the problem, so Joshua challenges, "What if there are a dozen circles?"
Tony gets to work on Joshua's problem of finding the maximum number of points of intersections where at least two of the twelve circles in a plane intersect. What is the answer to this problem?
Let $ABC$ be an acute triangle with altitudes $AA'$ and $BB'$ and orthocenter $H$. Let $C_0$ be the midpoint of the segment $AB$. Let $g$ be the line symmetric to the line $CC_0$ with respect to the angular bisector of $\angle ACB$. Let $h$ be the line symmetric to the line $HC_0$ with respect to the angular bisector of $\angle AHB$.
Show that the lines $g$ and $h$ intersect on the line $A'B'$.
The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.
The sides of a convex $200$-gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion.
Suppose the extensions of the red sides determine a regular $100$-gon, as do the extensions of the blue sides.
Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots,
\ \overline{A_{99}A_{199}}$ are concurrent.
[i]Proposed by: [b]Ankan Bhattacharya[/b][/i]
The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$, that is to say that $B$ touches both the top surface and the base and the sides. Calculate ratio volume $B :$ Volume $K$.
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
$\textbf{(A)} ~47 \qquad\textbf{(B)} ~94 \qquad\textbf{(C)} ~227 \qquad\textbf{(D)} ~471 \qquad\textbf{(E)} ~542$
Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. [hide=*]A tree is a connected graph with no cycles. A leaf is a vertex of degree 1[/hide]
[i]Vincent Huang[/i]
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?
A line $l$ passes through the center $O$ of an equilateral triangle $\Delta ABC$, which intersects $CA$ in $N$ and $BC$ in $M$. Prove that we can construct a triangle with $AM$,$BN$, and $MN$ such that the altitude to $MN$ (in this triangle) is constant when $l$ changes.
A beam of light shines from point $L$, reflects off a reflector at point $S$, and reaches point $D$ so that $\overline{SD}$ is perpendicular to $\overline{ML}$. Then $x$ is
[DIAGRAM NEEDED]
$ \mathrm{(A) \ } 13^\circ \qquad \mathrm{(B) \ } 26^\circ \qquad \mathrm {(C) \ } 32^\circ \qquad \mathrm{(D) \ } 58^\circ \qquad \mathrm{(E) \ } 64^\circ$
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.