Found problems: 85335
2019 ITAMO, 5
Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$
2012 IMO, 2
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2018 Bundeswettbewerb Mathematik, 1
Anja and Bernd take turns in removing stones from a heap, initially consisting of $n$ stones ($n \ge 2$). Anja begins, removing at least one but not all the stones. Afterwards, in each turn the player has to remove at least one stone and at most as many stones as removed in the preceding move. The player removing the last stone wins.
Depending on the value of $n$, which player can ensure a win?
2014 ASDAN Math Tournament, 13
Square $S_1$ is inscribed inside circle $C_1$, which is inscribed inside square $S_2$, which is inscribed inside circle $C_2$, which is inscribed inside square $S_3$, which is inscribed inside circle $C_3$, which is inscribed inside square $S_4$.
[center]<see attached>[/center]
Let $a$ be the side length of $S_4$, and let $b$ be the side length of $S_1$. What is $\tfrac{a}{b}$?
2014 Singapore Senior Math Olympiad, 2
Find the value of $\frac{2014^3-2013^3-1}{2013\times 2014}$.
$ \textbf{(A) }3\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }11 $
1996 Polish MO Finals, 2
Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$
2022 Cyprus JBMO TST, 3
If $a,b,c$ are positive real numbers with $abc=1$, prove that
(a) \[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca}\]
(b)\[2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}\]
2015 Thailand Mathematical Olympiad, 4
Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
2013 NIMO Problems, 8
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ-AX=6$, $BX-BZ=9$, $AY=12$, and $BY=5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.
[i]Proposed by Evan Chen[/i]
2017 CCA Math Bonanza, T9
Aida made three cubes with positive integer side lengths $a,b,c$. They were too small for her, so she divided them into unit cubes and attempted to construct a cube of side $a+b+c$. Unfortunately, she was $648$ blocks off. How many possibilities of the ordered triple $\left(a,b,c\right)$ are there?
[i]2017 CCA Math Bonanza Team Round #9[/i]
2008 JBMO Shortlist, 9
Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that:
$\frac{4a + p}{b}+\frac{4b + p}{a}$ and $ \frac{a^2}{b}+\frac{b^2}{a}$
are integers.
2017 Polish MO Finals, 6
Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that
$$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$
2009 Brazil Team Selection Test, 2
Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that
$$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$
2009 Korea - Final Round, 1
$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.
2020 BMT Fall, 2
There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the $38$ people already in the CBL.
2013 Tournament of Towns, 2
On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram
2008 Singapore Team Selection Test, 3
Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.
2013 India Regional Mathematical Olympiad, 6
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
2007 Stanford Mathematics Tournament, 8
Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?
1994 USAMO, 5
Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that \[ \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) \] for all integers $\, m \geq \sigma(S)$.
2023 Canada National Olympiad, 1
William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?
2008 District Olympiad, 1
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that
$$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$
Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $
1998 Chile National Olympiad, 2
Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.
a) Prove that the lines $ EF $ and $ AB $ are perpendicular.
b) Determine the locus of the point $ E $.
c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.
2011 Today's Calculation Of Integral, 681
Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$
[i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]
2006 Turkey Team Selection Test, 2
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?