Found problems: 1679
2008 IMC, 1
Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}\minus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$. Prove that $ x^{2k}\plus{}x^k\plus{}1$ divides $ x^{2n}\plus{}x^n\plus{}1$.
2003 China Team Selection Test, 3
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that:
\[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \]
(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?
1989 Kurschak Competition, 1
In the plane, two intersecting lines $a$ and $b$ are given, along with a circle $\omega$ that has no common points with these lines. For any line $\ell||b$, define $A=\ell\cap a$, and $\{B,C\}=\ell\cap \omega$ such that $B$ is on segment $AC$. Construct the line $\ell$ such that the ratio $\frac{|BC|}{|AB|}$ is maximal.
1954 AMC 12/AHSME, 28
If $ \frac{m}{n}\equal{}\frac{4}{3}$ and $ \frac{r}{t}\equal{}\frac{9}{14}$, the value of $ \frac{3mr\minus{}nt}{4nt\minus{}7mr}$ is:
$ \textbf{(A)}\ \minus{}5 \frac{1}{2} \qquad
\textbf{(B)}\ \minus{}\frac{11}{14} \qquad
\textbf{(C)}\ \minus{}1\frac{1}{4} \qquad
\textbf{(D)}\ \frac{11}{14} \qquad
\textbf{(E)}\ \minus{}\frac{2}{3}$
2025 Euler Olympiad, Round 1, 8
Let $S$ be the set of non-negative integer powers of $3$ and $5$, $S = \{1, 3, 5, 3^2, 5^2, \ldots \}$. For every $a$ and $b$ in $S$ satisfying $$ \left| \pi - \frac{a}{b} \right| < 0.1 $$ Find the minimum value of $ab$.
[i]Proposed by Irakli Shalibashvili, Georgia [/i]
1986 IMO Longlists, 59
Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$
[b](a)[/b] Prove that $ABCD$ and $A''B''C''D''$ are similar.
[b](b) [/b]The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.
2012 Math Prize For Girls Problems, 9
Bianca has a rectangle whose length and width are distinct primes less than 100. Let $P$ be the perimeter of her rectangle, and let $A$ be the area of her rectangle. What is the least possible value of $\frac{P^2}{A}$?
2000 Harvard-MIT Mathematics Tournament, 28
What is the smallest possible volume to surface ratio of a solid cone with height = $1$ unit?
2002 Turkey MO (2nd round), 2
Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.
2011 Purple Comet Problems, 20
Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter so that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of the circle is $\tfrac{17}{128}$. There are relatively prime positive integers m and n so that the ratio of $DE$ to $AB$ is $\tfrac{m}{n}.$ Find $m + n$.
2014 Contests, 2
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$
2004 Turkey MO (2nd round), 5
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
2005 India National Olympiad, 1
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$
than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
2010 Middle European Mathematical Olympiad, 10
Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 6)[/i]
1997 APMO, 3
Let $ABC$ be a triangle inscribed in a circle and let
\[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \]
where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that
\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \]
and that equality holds iff $ABC$ is an equilateral triangle.
2010 Danube Mathematical Olympiad, 2
Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively.
(a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$
(b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]
2013 ELMO Shortlist, 13
In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$.
[i]Proposed by Ray Li[/i]
2006 Tuymaada Olympiad, 4
For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$).
[i]Proposed by A. Golovanov[/i]
2010 Contests, 1
$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.
2015 Junior Balkan Team Selection Tests - Romania, 3
Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?
1957 Moscow Mathematical Olympiad, 365
(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$
(b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$
2011 AMC 10, 11
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $
1974 AMC 12/AHSME, 30
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $ R$ is the ratio of the lesser part to the greater part, then the value of \[ R^{[R^{(R^2\plus{}R^{\minus{}1})}\plus{}R^{\minus{}1}]}\plus{}R^{\minus{}1}\] is
$ \textbf{(A)}\ 2
\qquad \textbf{(B)}\ 2R
\qquad \textbf{(C)}\ R^{\minus{}1}
\qquad \textbf{(D)}\ 2\plus{}R^{\minus{}1}
\qquad \textbf{(E)}\ 2\plus{}R$
2025 Romania EGMO TST, P3
$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\dfrac{AM}{AO}$.
[i]Proposed by Fedir Yudin[/i]