Found problems: 85335
Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.
I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a [i]dime for a dozen[/i] pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend [b]all[/b] $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I [i]give my two cents[/i] to Bill, he then notes that he can instead spend [b]all[/b] $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible?
Note: Both times Bill must spend [b]all[/b] of his cents on pencil packages, i.e. have zero cents after either purchase.
Determine all integers $n \geq 2$ such that there exists a permutation $x_0, x_1, \ldots, x_{n - 1}$ of the numbers $0, 1, \ldots, n - 1$ with the property that the $n$ numbers
$$x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots + x_{n - 1}$$
are pairwise distinct modulo $n$.
Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.
Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a\cdot 21^b}$ is a positive integer.
[i]Proposed by Aidan Duncan[/i]
An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates $(0, 0), (p, 0), (p, q), (0, q)$ for some positive integers $p, q$. Show that there must exist two among them one of which is entirely contained in the other.
Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?
A monic polynomial is a polynomial whose highest degree coefficient is 1. Let $P(x)$ and $Q(x)$ be monic polynomial with real coefficients and $degP(x)=degQ(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions then $P(x+1)=Q(x-1)$ has a real solution
Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Solve the system of equations
$$\begin{cases} x+y+z+t=6
\\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8
\end{cases}$$
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$.
Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$.
Prove that $IE \cdot IK= IC \cdot IL$.
Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.
Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
Let $N$ be a positive integer. Let $R$ denote the smallest positive number that is the sum of $m$ terms $\sum^m_{i=1}{\pm \sqrt{a_i}}$, where each $a_i, i=1,\cdots, m$ is an integer not larger than $N$. Prove that \[R\le C\cdot N^{-m+\frac{3}{2}}\] for some positive real number $C$.
[i]Proposed by Navid[/i]
[i](Clarification: note that the constant is allowed to depend on $m$ but should be independent of $N$, i.e. the equation $R(m,N)\le C(m)\cdot N^{-m+\frac{3}{2}}$ should hold for all positive integers $N$)[/i]
Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
$
\textbf{(A) } \frac{1}{22,050} \qquad
\textbf{(B) } \frac{1}{21,000}\qquad
\textbf{(C) } \frac{1}{10,500}\qquad
\textbf{(D) } \frac{1}{2,100} \qquad
\textbf{(E) } \frac{1}{1,050}
$
Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$.
Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of $2.$
The positive numbers $a, b, c$ satisfy $4abc(a+b+c) = (a+b)^2(a+c)^2$. Prove that $a(a+b+c)=bc$.
[i]Proposed by Aaron Lin[/i]
A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of
\[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \]
can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$