Found problems: 85335
Let $D$ be a point on side $[BC]$ of $\triangle ABC$ such that $|AB|=|AC|$, $|BD|=6$ and $|DC|=10$. If the incircles of $\triangle ABD$ and $\triangle ADC$ touch side $[AD]$ at $E$ and $F$, respectively, what is$|EF|$?
$
\textbf{(A)}\ \dfrac{1}{\sqrt{2}}
\qquad\textbf{(B)}\ \dfrac{2}{\sqrt{3}}
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ \dfrac{9}{8}
\qquad\textbf{(E)}\ 2
$
Solve the equation $\sqrt{x} = \log_2 x$.
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.
A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards?
[asy]
size(300);
draw(circle((0,0),7));
path A=(1,2)--(1,-2)--(-1,-2)--(-1,2)--cycle;
filldraw(shift(7*left)*A,lightgray);
draw((-7,0)--(-7,5),EndArrow(size=21));
label(scale(1.5)*"A",(-8,2),2*N);
label(scale(1.5)*"B",(-8,0),2*W);
label(scale(1.5)*"C",(-7,-2),3*S);
label(scale(1.5)*"D",(-6,0),2*E);
[/asy]
(A) A
(B) B
(C) C
(D) D
(E) Remains vertical
It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$;
(a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$;
(b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.
Prove that there are in finitely many positive integers $n$ such that $\Delta = nr^2$, where $\Delta$ and $r$ are respectively the area and the inradius of a triangle with integer sides.
Find a positive integer $n$ such that there is a polygon with $n$ sides where each of its interior angles measures $177^o$
Let $ABCD$ be a quadrilateral with side lengths $AB = 2$, $BC = 3$, $CD = 5$, and $DA = 4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$?
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh
Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).
Let $ABC$ be a triangle, and $M_a, M_b, M_c$ be the midpoints of $BC, CA, AB$, respectively. Extend $M_bM_c$ so that it intersects $\odot (ABC)$ at $P$. Let $AP$ and $BC$ intersect at $Q$. Prove that the tangent at $A$ to $\odot(ABC)$ and the tangent at $P$ to $\odot (P QM_a)$ intersect on line $BC$.
(Li4)
What is the largest integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y$, and $z$?
Beatrix is going to place six rooks on a $6\times6$ chessboard where both the rows and columns are labelled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The [i]value[/i] of a square is the sum of its row number and column number. The [i]score[/i] of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Find the biggest non-integer $x$ such that $(x+2)^2 + (x+3)^3 + (x+4)^4 = 2$.
In triangle $ABC$ we have $AB=36$, $BC=48$, $CA=60$. The incircle of $ABC$ is centered at $I$ and touches $AB$, $AC$, $BC$ at $M$, $N$, $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $MAN$ and $KID$ intersects lines $AB$ and $AC$ at $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.
[i]Proposed by Evan Chen[/i]
A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number of cubic feet of dirt needed to form the pile can be written as $m + n \pi$ where $m$ and $n$ are positive integers. Find $m + n.$
Call a set of integers [i]spacy[/i] if it contains no more than one out of any three consecutive integers. How many subsets of $ \{1,2,3,\ldots,12\},$ including the empty set, are spacy?
$ \textbf{(A)}\ 121 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 125 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 129$
Find all functions $f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}$ such that for any two rational numbers $x$ and $y$ the following conditions hold
[list]
[*] $f(x+y)\leq f(x)+f(y)$,
[*]$f(xy)=f(x)f(y)$,
[*]$f(2)=1/2$.
[/list]
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$
Prove that circles which have sides of a convex quadrilateral as diameters cover its interior. (Convex polygon is the one which contains with any two points the whole segment, joining them).
Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ .
Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$ \textbf{(A)}\ 72\qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ \text{not uniquely determined} $
let $ H $ the orthocenter of the triangle $ ABC $
pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $
we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $
also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture
pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $
let $ d_1 \cap d_2 = Y $
prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$