Found problems: 85335
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.
Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]
and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?
[i]Carl Schildkraut and Milan Haiman[/i]
Let $[ABCD]$ be a square, $M$ a point on the segment $[AD]$, and $N$ a point on the segment $[DC]$ such that $B\hat{M}A = N\hat{M}D = 60^{\circ}$. Calculate $M\hat{B}N$.
Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle
Find integer solutions of $x^3+y^3-2xy+x+y+2=0$
$ABC$ is an equilateral triangle, and $ADEF$ is a square. If $D$ lies on side $AB$ and $E$ lies on side $BC$, what is the ratio of the area of the equilateral triangle to the area of the square?
A polynomial
\[ P(x) \equal{} c_{2004}x^{2004} \plus{} c_{2003}x^{2003} \plus{} ... \plus{} c_1x \plus{} c_0
\]has real coefficients with $ c_{2004}\not \equal{} 0$ and $ 2004$ distinct complex zeroes $ z_k \equal{} a_k \plus{} b_ki$, $ 1\leq k\leq 2004$ with $ a_k$ and $ b_k$ real, $ a_1 \equal{} b_1 \equal{} 0$, and
\[ \sum_{k \equal{} 1}^{2004}{a_k} \equal{} \sum_{k \equal{} 1}^{2004}{b_k}.
\]Which of the following quantities can be a nonzero number?
$ \textbf{(A)}\ c_0 \qquad \textbf{(B)}\ c_{2003} \qquad \textbf{(C)}\ b_2b_3...b_{2004} \qquad \textbf{(D)}\ \sum_{k \equal{} 1}^{2004}{a_k} \qquad \textbf{(E)}\ \sum_{k \equal{} 1}^{2004}{c_k}$
Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.
Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.
Carrie, Miranda, Charlotte, and Samantha are sitting at a table with $5$ numbered chairs (numbered $1$ through $5$). One chair is left open for Big, should he decide to join the four for lunch. In how many distinct ways can the four women occupy the table?
You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ [i]good[/i] if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane.
[i]Proposed by Anton Trygub[/i]
In the $(x,y)-$plane, if $R$ is the set of points inside and on a convex polygon, let $D(x,y)$ be the distance from $(x,y)$ to the nearest point of $R.$
(a) Show that there exists constants $a,b,c,$ independent of $R$, such that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-D(x,y)} dxdy =a+bL+cA,$$ where $L$ is the perimeter of $R$ and $A$ is the area of $R.$
(b) Find the values of $a,b$ and $c.$
The value(s) of $y$ for which the following pair of equations \[x^2+y^2-16=0\text{ and }x^2-3y+12=0\] may have a real common solution, are
$\textbf{(A) }4\text{ only}\qquad\textbf{(B) }-7,~4\qquad\textbf{(C) }0,~4\qquad\textbf{(D) }\text{no }y\qquad \textbf{(E) }\text{all }y$
Find all positive integers $ n$ such $ 20n\plus{}2$ can divide $ 2003n \plus{} 2002.$
A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.
Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).
Find the largest number $\lambda$ such that $a^2+b^2+c^2+d^2 \geq ab + \lambda bc + cd$ for all real numbers $a,b,c,d$
A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$.
[asy]
size(200);
pathpen=black+linewidth(0.65);pointpen=black;
pair A=(0,0),B=(12,0),C=(4,5);
D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12);
MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */
MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N);
MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW);
MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]
There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
Peter has several $100$ ruble notes and no other money. He starts buying books; each book costs a positive integer number of rubles, and he gets change in $1$ ruble coins. Whenever Peter is buying an expensive book for $100$ rubles or higher he uses only $100$ ruble notes in the minimum necessary number. Whenever he is buying a cheap one (for less than $100$ rubles) he uses his coins if he has enough, otherwise using a $100$ ruble note.
When the $100$ ruble notes have come to the end, Peter has expended exactly a half of his money. Is it possible that he has expended $5000$ rubles or more?
(Tatiana Kazitsina)
Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.
Prove that there do not exist eleven primes, all less than $20000$, which form an arithmetic progression.
Let $a$ be a real number. Find all real-valued functions $f$ such that
$$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$
when constants of integration are suitably chosen.