Found problems: 85335
1992 IMO Shortlist, 14
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
1982 Czech and Slovak Olympiad III A, 4
In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.
2011 IMO Shortlist, 6
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$.
[i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]
2004 APMO, 5
Prove that the inequality \[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\] holds for all positive reals $a$, $b$, $c$.
2023 MOAA, 6
Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$.
[i]Proposed by Anthony Yang[/i]
2016 Bosnia and Herzegovina Team Selection Test, 5
Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear.
.
2010 N.N. Mihăileanu Individual, 2
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality
$$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$
for any real number $ x. $ Prove that $ f=0. $
[i]Nelu Chichirim[/i]
1968 Vietnam National Olympiad, 2
$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$
1971 IMO, 2
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
2025 Harvard-MIT Mathematics Tournament, 17
Let $f$ be a quadratic polynomial with real coefficients, and let $g_1, g_2, g_3, \ldots$ be a geometric progression of real numbers. Define $a_n=f(n)+g_n.$ Given that $a_1, a_2, a_3, a_4,$ and $a_5$ are equal to $1, 2, 3, 14,$ and $16,$ respectively, compute $\tfrac{g_2}{g_1}.$
2017 JBMO Shortlist, A4
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1994 Tournament Of Towns, (432) 2
Prove that one can construct two triangles from six edges of an arbitrary tetrahedron.
(VV Proizvolov)
2016 SEEMOUS, Problem 1
SEEMOUS 2016 COMPETITION PROBLEMS
2021 Vietnam National Olympiad, 3
Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$.
a) Prove that $AL$ is perpendicular to $EF$.
b) Let $AL$ intersects $EF$ at $M$, the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$, $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.
2006 South East Mathematical Olympiad, 2
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
2008 BAMO, 3
A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles.
(A triangle is isosceles if it has at least two sides the same length.)
2002 AMC 10, 8
Suppose July of year $ N$ has five Mondays. Which of the following must occur five times in August of year $ N$? (Note: Both months have $ 31$ days.)
$ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Thursday} \qquad
\textbf{(E)}\ \text{Friday}$
2014 Sharygin Geometry Olympiad, 22
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?
2013 Chile National Olympiad, 2
Hannibal and Clarice are still at a barbecue and there are three anticuchos left, each of which it has $10$ pieces. Of the $30$ total pieces, there are $29$ chicken and one meat, the which is at the bottom of one of the anticuchos. To decide who to stay with the piece of meat, they decide to play the following game: they alternately take out a piece of one of the anticuchos (they can take only the outer pieces) and whoever wins the game manages to remove the piece of meat. Clarice decides if she starts or Hannibal starts. What should she decide?
1990 Baltic Way, 8
It is known that for any point $P$ on the circumcircle of a triangle $ABC$, the orthogonal projections of $P$ onto $AB,BC,CA$ lie on a line, called a [i]Simson line[/i] of $P$. Show that the Simson lines of two diametrically opposite points $P_1$ and $P_2$ are perpendicular.
2017 India IMO Training Camp, 2
Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.
2004 Brazil Team Selection Test, Problem 2
An integer $n\ge2$ is called [i]amicable[/i] if there exists subsets $A_1,A_2,\ldots,A_n$ of the set $\{1,2,\ldots,n\}$ such that
(i) $i\notin A_i$ for any $i=1,2,\ldots,n$,
(ii) $i\in A_j$ for any $j\notin A_i$, for any $i\ne j$
(iii) $A_i\cap A_j\ne\emptyset$ for any $i,j\in\{1,2,\ldots,n\}$
(a) Prove that $7$ is amicable.
(b) Prove that $n$ is amicable if and only if $n\ge7$.
2021 Saudi Arabia BMO TST, 3
Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a^5 - a^2 +3)(b^5 - b^2 +3)(c^5 - c^2 +3)\ge (a+b+c)^3$$
1997 Tournament Of Towns, (540) 5
In a game, the first player paints a point on the plane red; the second player paints 10 uncoloured points on the plane green; then the first player paints an uncoloured point on the plane red; the second player paints 10 uncoloured points on the plane green; and so on. The first player wins if there are three red points which form an equilateral triangle. Can the second player prevent the first player from winning?
(A Kanel)
2011 NIMO Summer Contest, 13
For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression
\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]
[i]Proposed by Lewis Chen
[/i]