Found problems: 85335
2011 Princeton University Math Competition, A6 / B7
For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?
1997 Baltic Way, 6
Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and
\[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]
2007 Vietnam Team Selection Test, 2
Let $ABC$ be an acute triangle with incricle $(I)$. $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$, similary we have $B_{1},C_{1}$.
a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$.
b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$.
Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$.
1998 IMO Shortlist, 2
Let $r_{1},r_{2},\ldots ,r_{n}$ be real numbers greater than or equal to 1. Prove that
\[ \frac{1}{r_{1} + 1} + \frac{1}{r_{2} + 1} + \cdots +\frac{1}{r_{n}+1} \geq \frac{n}{ \sqrt[n]{r_{1}r_{2} \cdots r_{n}}+1}. \]
2015 Canadian Mathematical Olympiad Qualification, 2
A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.
2010 Purple Comet Problems, 7
$x$ and $y$ are positive real numbers where $x$ is $p$ percent of $y$, and $y$ is $4p$ percent of $x$. What is $p$?
2019 Harvard-MIT Mathematics Tournament, 2
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?
2008 IMO Shortlist, 7
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality
\[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\]
holds. Determine all cases of equality.
[i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]
2021 JHMT HS, 9
Define a sequence $\{ a_n \}_{n=0}^{\infty}$ by $a_0 = 1,$ $a_1 = 8,$ and $a_n = 2a_{n-1} + a_{n-2}$ for $n \geq 2.$ The infinite sum
\[ \sum_{n=1}^{\infty} \int_{0}^{2021\pi/14} \sin(a_{n-1}x)\sin(a_nx)\,dx \]
can be expressed as a common fraction $\tfrac{p}{q}.$ Compute $p + q.$
2012 Tournament of Towns, 2
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
2010 LMT, 5
Evaluate $2010^2-2009\cdot2011.$
2019 Harvard-MIT Mathematics Tournament, 4
Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.
1990 IMO Shortlist, 4
Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.
2013 China Second Round Olympiad, 1
For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.
1993 AIME Problems, 8
Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.
2016 Polish MO Finals, 5
There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions:
$1$. $a< \frac{p}{q} < \frac{r}{s} < b$.
$2.$ $p^2+q^2=r^2+s^2$.
2008 Sharygin Geometry Olympiad, 4
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.
Kyiv City MO Juniors 2003+ geometry, 2009.89.5
A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.
2014 Iran Geometry Olympiad (senior), 3:
Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $.
Author:Ali zo'alam , Iran
2006 Swedish Mathematical Competition, 3
A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.
2022 AMC 10, 3
How many three-digit positive integers have an odd number of even digits?
$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$
2015 Moldova Team Selection Test, 1
Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.
2007 Postal Coaching, 2
Let $a_1, a_2, a_3$ be three distinct real numbers. Define
$$\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\
b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\
b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}$$
Prove that $$1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)$$
When does equality hold?
2011 Saudi Arabia Pre-TST, 1.4
Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC = 40^o$. Points $S$ and $T$ lie on the sides $AB$ and $BC$, such that $\angle BAT = \angle BCS = 10^o$. Lines $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.
2013 Bundeswettbewerb Mathematik, 3
Let $ABCDEF$ be a convex hexagon whose vertices lie on a circle. Suppose that $AB\cdot CD\cdot EF = BC\cdot DE\cdot FA$. Show that the diagonals $AD, BE$ and $CF$ are concurrent.