Found problems: 85335
Prove that a section of a cube by a plane cannot be a regular pentagon.
There are $12$ monsters in a plane. Each monster is capable of spraying fire in a $30$-degree cone. Prove that monsters can destroy the plane.
Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.
How many ordered triples of integers $(x,y,z)$ satisfy \[36x^2+100y^2+225z^2=12600?\]
[i]Proposed by Bill Fei and Mahith Gottipati [/i]
A triangle has side lengths $a, b, c$. Prove that
$$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$
Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.
We define the sequence
$x_1=n,y_1=1,x_{i+1}=[\frac{x_i+y_i}{2}],y_{i+1}=[\frac{n}{x_{i+1}} ]$.
Prove that $min\{ x_1, x_2, ..., x_n\}=[\sqrt{n}]$ .
Prove that there exists a family $\mathfrak F=\{A_1,A_2,\ldots,A_r\}$ of $m$-element subsets of a given set $\{b_1,b_2,\ldots,b_n\}$ of $n$ elements such that
(i) $\left|A_i\cap A_j\right|\le m-2$ for all $A_i,A_j\in\mathfrak F$ with $i\ne j$, and
(ii) $r\ge\left\lfloor\frac1n\binom nm\right\rfloor$
Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
\[\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.\]
Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.
The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common.
Let $A,B\in\mathbb{C}^{n\times n}$ with $\rho(AB - BA) = 1$. Show that $(AB - BA)^2 = 0$.
At a certain school, the ratio of boys to girls is $1:3$. Suppose that:
$\bullet$ Every boy has most $2013$ distinct girlfriends.
$\bullet$ Every girl has at least $n$ boyfriends.
$\bullet$ Friendship is mutual.
Compute the largest possible value of $n$.
[i]Proposed by Evan Chen[/i]
The following number is the product of the divisors of $n$.
$$46, 656, 000, 000$$
What is $n$?
Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call [i]configuration [/i] of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$, it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.
The polygon(s) formed by $y=3x+2$, $y=-3x+2$, and $y=-2$, is (are):
$ \textbf{(A) }\text{An equilateral triangle}\qquad\textbf{(B) }\text{an isosceles triangle} \qquad\textbf{(C) }\text{a right triangle} \qquad$
$\textbf{(D) }\text{a triangle and a trapezoid}\qquad\textbf{(E) }\text{a quadrilateral} $
Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)
S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt));
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,0)--(1,6));
draw((7,1)--(1,1));
draw((6,7)--(6,1));
draw((0,6)--(6,6));[/asy]$ \textbf{(A)}\ \ 49 \qquad \textbf{(B)}\ \ 64 \qquad \textbf{(C)}\ \ 100 \qquad \textbf{(D)}\ \ 121 \qquad \textbf{(E)}\ \ 196$
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
$\text{(A)}\ \text{1 hr} \qquad \text{(B)}\ \text{1 hr 10 min} \qquad \text{(C)}\ \text{1 hr 20 min} \qquad \text{(D)}\ \text{1 hr 40 min} \qquad \text{(E)}\ \text{2 hr}$
How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.