Found problems: 85335
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$
For positive integers $k$ and $n$, express the number of permutation $P=x_1x_2...x_{2n}$ consisting of $A$ and $B$ that satisfies all three of the following conditions, using $k$ and $n$.
$ $ $ $ $(i)$ $A, B$ appear exactly $n$ times respectively in $P$.
$ $ $ $ $(ii)$ For each $1\le i\le n$, if we denote the number of $A$ in $x_1,x_2,...,x_i$ as $a_i$ $,$ then $\mid 2a_i -i\mid \le 1$.
$ $ $ $ $(iii)$ $AB$ appears exactly $k$ times in $P$. (For example, $AB$ appears 3 times in $ABBABAAB$)
Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that
\[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\]
and $ \text{deg}P<\text{deg}{Q}.$
Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?
Determine all real constants $t$ such that whenever $a$, $b$ and $c$ are the lengths of sides of a triangle, then so are $a^2+bct$, $b^2+cat$, $c^2+abt$.
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there?
Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)
Given a trapezoid $ABCD$, with $AB\parallel CD$. Lines $AC$ and $BD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. It turns out that the circle with diameter $EF$ is tangent to the midline of the trapezoid. Prove that there exists a square such that there is a mutual correspondence between all six lines containing pairs of its vertices, and points $A$, $B$, $C$, $D$, $E$, and $F$: each line corresponds to a point lying on it.
[i]Proposed by V. Mokin[/i]
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
[i]Author: Anderson Wang[/i]
Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.
Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold
[list]
[*] $a+b \in F$, and
[*] $\gcd(P(a),P(b))=1$.
[/list]
Prove that $P(x)$ is a constant polynomial.
In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$.
[i]Proposed by Andrew Wen[/i]
Numbers $1,2,3...,100$ are written on a board. $A$ and $B$ plays the following game: They take turns choosing a number from the board and deleting them. $A$ starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.
Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations
$$\frac{x}{\sqrt{x^2 + y^2}}-\frac{1}{x}= 7 \,\,\, \text{and} \,\,\, \frac{y}{\sqrt{x^2 + y^2}}+\frac{1}{y}=4 $$
Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$.
Alexandru Gırban
Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose $AD, BE,CF$ are concurrent at $P$. If $PF/PC =2/3, PE/PB = 2/7$ and $PD/PA = m/n$, where $m, n$ are positive integers with $gcd(m, n) = 1$, find $m + n$.
Let $\mathcal{F}$ be a finite collection of open discs in $\mathbb{R}^2$ whose union contains a set $E\subseteq \mathbb{R}^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots,D_n$ in $\mathcal{F}$ such that \[E\subseteq\cup_{j=1}^n 3D_j.\] Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.
[i]Superchess[/i] is played on on a $12 \times 12$ board, and it uses [i]superknights[/i], which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every other cell of a superchessboard exactly once and return to its starting cell ?
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$.
Suppose that $AM$ intersects the incircle at $K,L$.
We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$.
Prove that $BP=CQ$.
Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$?
[i]Proposed by Sammy Luo[/i]
$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$
The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term.
${{ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt[7]{2} \qquad\textbf{(C)}\ \sqrt[8]{2} \qquad\textbf{(D)}\ \sqrt[9]{2} }\qquad\textbf{(E)}\ \sqrt[10]{2} } $