Found problems: 85335
2017 Polish MO Finals, 5
Point $M$ is the midpoint of $BC$ of a triangle $ABC$, in which $AB=AC$. Point $D$ is the orthogonal projection of $M$ on $AB$. Circle $\omega$ is inscribed in triangle $ACD$ and tangent to segments $AD$ and $AC$ at $K$ and $L$ respectively. Lines tangent to $\omega$ which pass through $M$ cross line $KL$ at $X$ and $Y$, where points $X$, $K$, $L$ and $Y$ lie on $KL$ in this specific order. Prove that points $M$, $D$, $X$ and $Y$ are concyclic.
JOM 2015 Shortlist, A9
Let \(2n\) positive reals \(a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n\) satisfy \(a_{i+1}\ge 2a_i\) and \(b_{i+1} \le b_i\) for \(1\le i\le n-1\). Find the least constant \(C\) that satisfy: \[\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}\] and determine all equality case with that constant \(C\).
2015 Turkmenistan National Math Olympiad, 4
Find the max and minimum without using dervivate:
$\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$
2008 Moldova National Olympiad, 9.3
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?
2009 Harvard-MIT Mathematics Tournament, 8
Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.
2007 Tournament Of Towns, 7
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."
[list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot?
[b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to
the empty spot?[/list]
2014 Sharygin Geometry Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.
(V. Yasinsky)
1956 AMC 12/AHSME, 50
In triangle $ ABC$, $ \overline{CA} \equal{} \overline{CB}$. On $ CB$ square $ BCDE$ is constructed away from the triangle. If $ x$ is the number of degrees in angle $ DAB$, then
$ \textbf{(A)}\ x\text{ depends upon triangle }ABC \qquad\textbf{(B)}\ x\text{ is independent of the triangle}$
$ \textbf{(C)}\ x\text{ may equal angle }CAD \qquad\textbf{(D)}\ x\text{ can never equal angle }CAB$
$ \textbf{(E)}\ x\text{ is greater than }45^{\circ}\text{ but less than }90^{\circ}$
2018 Azerbaijan BMO TST, 1
Find all positive integers $(x,y)$ such that
$x^2+y^2=2017(x-y)$
1952 AMC 12/AHSME, 35
With a rational denominator, the expression $ \frac {\sqrt {2}}{\sqrt {2} \plus{} \sqrt {3} \minus{} \sqrt {5}}$ is equivalent to:
$ \textbf{(A)}\ \frac {3 \plus{} \sqrt {6} \plus{} \sqrt {15}}{6} \qquad\textbf{(B)}\ \frac {\sqrt {6} \minus{} 2 \plus{} \sqrt {10}}{6} \qquad\textbf{(C)}\ \frac {2 \plus{} \sqrt {6} \plus{} \sqrt {10}}{10}$
$ \textbf{(D)}\ \frac {2 \plus{} \sqrt {6} \minus{} \sqrt {10}}{6} \qquad\textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 557
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
2012 AMC 10, 2
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $
2006 Baltic Way, 5
An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:
$\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$;
$\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$.
The professor claims in his book that
$1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$.
$2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$.
Which of these claims follow from the stated axioms?
2005 Bulgaria Team Selection Test, 3
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
2011 Sharygin Geometry Olympiad, 14
In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.
2011 Iran MO (3rd Round), 1
We define the recursive polynomial $T_n(x)$ as follows:
$T_0(x)=1$
$T_1(x)=x$
$T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$.
[b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$.
[b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$.
[i]Proposed by Morteza Saghafian[/i]
2013 BAMO, 4
Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$.
Example:
[img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]
2005 JBMO Shortlist, 7
Let $ABCD$ be a parallelogram. $P \in (CD), Q \in (AB)$, $M= AP \cap DQ$, $N=BP \cap CQ$, $ K=MN \cap AD$, $L= MN \cap BC$. Prove that $BL=DK$.
2007 Swedish Mathematical Competition, 3
Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that
\[
\cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc}
\]
1989 APMO, 4
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least
\[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \]
triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.
2004 AMC 10, 18
A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 49\qquad
\textbf{(E)}\ 81$
2010 ELMO Shortlist, 2
Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\]
[i]Timothy Chu.[/i]
2003 AMC 12-AHSME, 13
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?
$ \textbf{(A)}\ 2: 1 \qquad
\textbf{(B)}\ 3: 1 \qquad
\textbf{(C)}\ 4: 1 \qquad
\textbf{(D)}\ 16: 3 \qquad
\textbf{(E)}\ 6: 1$
2022 AMC 10, 14
Suppose that $S$ is a subset of $\{1, 2, 3,...,25\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S$. What is the maximum number of elements $S$ may contain?
$\textbf{(A) }12 \qquad \textbf{(B) }13 \qquad \textbf{(C) }14 \qquad \textbf{(D) }15 \qquad \textbf{(E) }16$
2018 China Team Selection Test, 1
Given a triangle $ABC$. $D$ is a moving point on the edge $BC$. Point $E$ and Point $F$ are on the edge $AB$ and $AC$, respectively, such that $BE=CD$ and $CF=BD$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ intersects at another point $P$ other than $D$. Prove that there exists a fixed point $Q$, such that the length of $QP$ is constant.