Found problems: 85335
Let $n>k$ and let $A_1,\ldots,A_k$ be real $n\times n$ matrices of rank $n-1$. Prove that
$$A_1\cdots A_k\ne0.$$
$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $
$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$).
Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
On the sides $AB,BC,CD$ and $DA$ of a unit square $ABCD$ points $P,Q,R$ and $S$ are chosen respectively. It turned out that the perimeter of $PQRS$ is $2\sqrt{2}$. Find the sum of perpendiculars from $A,B,C,D$ to $SP,PQ,QR,RS$ respectively.
Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.
Let $u$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y )$ satisfy all of the following conditions:
$\text{(i) }\frac{a}{2}\le x\le 2a\qquad\text{(ii) }\frac{a}{2}\le y\le 2a\qquad\text{(iii) }x+y\ge a \\ \\ \qquad\text{(iv) }x+a\ge y\qquad \text{(v) }y+a\ge x$
The boundary of set $S$ is a polygon with
$\textbf{(A) }3\text{ sides}\qquad\textbf{(B) }4\text{ sides}\qquad\textbf{(C) }5\text{ sides}\qquad\textbf{(D) }6\text{ sides}\qquad \textbf{(E) }7\text{ sides}$
Suppose that we are given an $n$-gon of which all sides have the same length, and of which all the vertices have rational coordinates. Prove that $n$ is even.
Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.
Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]
What is the measure of the largest convex angle formed by the hour and minute hands of a clock between $1:45$ PM and $2:40$ PM, in degrees? Convex angles always have a measure of less than $180$ degrees.
$f(x)=\frac{4^x}{4^x+2}$, then $f(\frac{1}{1001})+f(\frac{2}{1001})+\cdots+f(\frac{1000}{1001})=$________.
$$\int_0^\infty \frac{1}{(x^2+4)^{5/2}}\,dx$$
[i]Proposed by Connor Gordon[/i]
Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.
Let each sequence of capital Bulgarian letters be a word (Note that the Bulgarian alphabet consists of 30 letters). We say that a word is [i]calm[/i], if there is no sequence of the letter $P$ in it (two letters $P$ next to each other). Find a clear expression for $f(n)$ (closed formula), which represents the number of calm words with length $n$.
Example: [i]РНТ[/i] and [i]ТТРВ[/i] are calm, but [i]ЖГРР[/i] and [i]ЖРРРП[/i] aren’t.
In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.
$O$ is a circumcircle of $ABC$ and $CO$ meets $AB$ at $P$, and $BO$ meets $AC$ at $Q$. Show that $BP=PQ=QC$ if and only if $\angle A=60^{\circ}$.
Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$.
[img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.
Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles
[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$.
[b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$.
[b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$