Found problems: 85335
Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.
In a town there are four pubs, $A,B,C,D$, and any two of them are connected to each other except $A$ and $D$. A drunkard wanders about the pubs starting with $A$ and, after having a drink, goes to any of the pubs directly connected, with equal probability.
(a) What is the probability that the drunkard is at $C$ at its fifth drink?
(b) Where is the drunkard most likely to be after $n$ drinks ($n > 5$)?
A sequence ($a_n$) satisfies the following conditions:
(i) For each $m \in N$ it holds that $a_{2^m} = 1/m$.
(ii) For each natural $n \ge 2$ it holds that $a_{2n-1}a_{2n} = a_n$.
(iii) For all integers $m,n$ with $2m > n \ge 1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$.
Determine $a_{2000}$. You may assume that such a sequence exists.
$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.
Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac mn$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$
In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
We define a sequence $x_1 = \sqrt{3}, x_2 =-1, x_3 =2 - \sqrt{3},$ and for all $n \geq 4$
$$(x_n + x_{n-3})(1 - x^2_{n-1}x^2_{n-2}) = 2x_{n-1}(1 + x^2_{n-2}).$$
Suppose $m$ is the smallest positive integer for which $x_m$ is undefined. Compute $m.$
The positive integers $ a, b, c $ are the lengths of the sides of a right triangle. Prove that $abc$ is divisible by $60$.
Let $ ABCD$ be a parallelogram and let $ \overrightarrow{AA^\prime}$, $ \overrightarrow{BB^\prime}$, $ \overrightarrow{CC^\prime}$, and $ \overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ ABCD$. If $ AA^\prime \equal{} 10$, $ BB^\prime \equal{} 8$, $ CC^\prime \equal{} 18$, $ DD^\prime \equal{} 22$, and $ M$ and $ N$ are the midpoints of $ \overline{A^{\prime}C^{\prime}}$ and $ \overline{B^{\prime}D^{\prime}}$, respectively, then $ MN \equal{}$
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 3\qquad
\textbf{(E)}\ 4$
Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times?
[i]Proposed by Ethan Tan[/i]
Let $ABC$ be a triangle, $I$ its incenter and $\omega$ its incircle. Let $D$,$E$ and $F$ be the points of tangency of $\omega$ with $BC$,$AC$ and $AB$, respectively and $M$,$N$ and $P$ be the midpoints of $BC$, $AC$ and $AB$. Let $D'$ be the second intersection of $DI$ with $\omega$, $Q$ the intersection of $DI$ with $EF$ and $U \ne Q$ be the intersection of $(AD'Q)$ with $(DMQ)$. Suppose that $U$ lies on the circumcircle of $BDF$. Prove that $PN, AM, UF$ concur.
Country $A$ uses a currency known as the shell. The nation uses only two coins, each worth a whole number of shells. The largest amount of shell not obtainable using a combination of these two coins is $215$. Find the number of possible pairs of values these two coins could have. (a value of $15$ and $4$ is the same as having a $4$ and $15$)
$\textbf{(A) } 6\qquad\textbf{(B) } 7\qquad\textbf{(C) } 8\qquad\textbf{(D) } 9\qquad\textbf{(E) } 10$
Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.
Triangle $ABC$ has $AB=13$, $BC=14$, and $CA=15$. Let $\omega_A$, $\omega_B$ and $\omega_C$ be circles such that $\omega_B$ and $\omega_C$ are tangent at $A$, $\omega_C$ and $\omega_A$ are tangent at $B$, and $\omega_A$ and $\omega_B$ are tangent at $C$. Suppose that line $AB$ intersects $\omega_B$ at a point $X \neq A$ and line $AC$ intersects $\omega_C$ at a point $Y \neq A$. If lines $XY$ and $BC$ intersect at $P$, then $\tfrac{BC}{BP} = \tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Find $100m+n$.
[i]Proposed by Michael Ren[/i]
Find all function $f:\mathbb{R}\to \mathbb{R}$ such that
\[f(x)f(y)=f(xy-1)+yf(x)+xf(y)\]
for all $x,y \in \mathbb{R}$
Let $ n \geq 2$ be an integer, let $ S$ be a set of $ n$ elements, and let $ A_i , \; 1\leq i \leq m$, be distinct subsets of $ S$ of size at least $ 2$ such that \[ A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ .\] Show that $ m \leq 2^{n\minus{}1}\minus{}1$.
[i]P. Erdos[/i]
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).
[i]Evan O'Dorney and Victor Wang[/i]
Consider the addition problem:
\begin{tabular}{ccccc}
&C&A&S&H\\
+&&&M&E\\
\hline
O&S&I&D&E
\end{tabular}
where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent
the same digit.) How many ways are there to assign values to the letters so that the addition problem
is true?
Define the operation $\bigoplus$ on the set of real numbers such that
\[x \bigoplus y = x+y-xy \qquad \forall x,y \in \mathbb R.\]
Prove that this operation is associative.
Lukas Nepomuceno draws 5 congruent circles equally spaced around a 6th, and colors all of them 1 of 3 colors. Assume that rotations and reflections of colorings are indistinguishable. How many distinct colorings are there?
[i]2022 CCA Math Bonanza Lightning Round 4.4[/i]
If $0<a_1< \cdots < a_n$, show that the following equation has exactly $n$ roots.
$$ \frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+ \frac{a_3}{a_3-x}+ \cdots + \frac {a_n}{a_n - x} = 2015$$
Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card.
[i]E. Lakshtanov[/i]
Given $A(-2,2)$, and $B$ is a moving point on ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$. $F$ is the left focal point of the ellipse, find the coordinate of $B$ when $|AB|+\frac{5}{3}|BF|$ takes its minumum value.
Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.