Found problems: 85335
2021 Brazil Team Selection Test, 4
Find all positive integers $n$ with the folowing property: for all triples ($a$,$b$,$c$) of positive real there is a triple of non negative integers ($l$,$j$,$k$) such that $an^k$, $bn^j$ and $cn^l$ are sides of a non degenate triangle
2012 Kazakhstan National Olympiad, 2
Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that:
$a)$ Points $M,N,A$ and $B$ lie on a circle
$b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$
2000 Manhattan Mathematical Olympiad, 3
Suppose one has an unlimited supply of identical tiles in the shape of a right triangle
[asy]
draw((0,0)--(3,0)--(3,2)--(0,0));
label("$A$",(0,0),SW);
label("$B$",(3,0),SE);
label("$C$",(3,2),NE);
size(100);
[/asy]
such that, if we measure the sides $AB$ and $AC$ (in inches) their lengths are integers. Prove that one can pave a square completely (without overlaps) with a number of these tiles, exactly when $BC$ has integer length.
Gheorghe Țițeica 2024, P4
A factorization of a positive integers is a way of writing it as a product of positive integers greater than $1$. Two factorizations are considered the same if they only differ in the order of terms in the product. For instance, $18$ has $4$ different factorizations: $18, 2\cdot 9, 3\cdot 6$ and $ 2\cdot 3\cdot 3$. For a positive integer $n$ we denote by $f(n)$ the number of distinct factorizations of $n$. By convention $f(1)=1$. Prove that $f(n)\leq n$ for all positive integers $n$.
2006 Germany Team Selection Test, 1
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
2015 Romania Team Selection Tests, 1
Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$, situated inside triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.
2013 BMT Spring, 8
A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.
2015 Turkey MO (2nd round), 2
$x$, $y$ and $z$ are real numbers where the sum of any two among them is not $1$. Show that, \[ \dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}\]Find all triples $(x,y,z)$ of real numbers satisfying the equality case.
2010 Hanoi Open Mathematics Competitions, 10
Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$
2009 Abels Math Contest (Norwegian MO) Final, 4b
Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.
Indonesia MO Shortlist - geometry, g10
It is known that circle $\Gamma_1(O_1)$ has center at $O_1$, circle $\Gamma_2(O_2)$ has center at $O_2$, and both intersect at points $C$ and $D$. It is also known that points $P$ and $Q$ lie on circles $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$, respectively. ). A line $\ell$ passes through point $D$ and intersects $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$ at points $A$ and $B$, respectively. The lines $PD$ and $AC$ meet at point $M$, and the lines $QD$ and $BC$ meet at point $N$. Let $O$ be center outer circle of triangle $ABC$. Prove that $OD$ is perpendicular to $MN$ if and only if a circle can be found which passes through the points $P, Q, M$ and $N$.
2022 Polish MO Finals, 6
A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$.
2016 AMC 8, 4
When Cheenu was a boy he could run $15$ miles in $3$ hours and $30$ minutes. As an old man he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?
$\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$
1993 Rioplatense Mathematical Olympiad, Level 3, 6
Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.
1957 AMC 12/AHSME, 15
The table below shows the distance $ s$ in feet a ball rolls down an inclined plane in $ t$ seconds.
\[ \begin{tabular}{|c|c|c|c|c|c|c|}
\hline
t & 0 & 1 & 2 & 3 & 4 & 5\\
\hline
s & 0 & 10 & 40 & 90 & 160 & 250\\
\hline
\end{tabular}
\]
The distance $ s$ for $ t \equal{} 2.5$ is:
$ \textbf{(A)}\ 45\qquad
\textbf{(B)}\ 62.5\qquad
\textbf{(C)}\ 70\qquad
\textbf{(D)}\ 75\qquad
\textbf{(E)}\ 82.5$
2023 BMT, 5
Let $p$, $q$, and $r$ be the three roots of the polynomial $x^3 -2x^2 + 3x - 2023$. Suppose that the polynomial $x^3 + Bx^2 +Mx + T$ has roots $p + q$, $p + r$, and $q + r$ for real numbers $B$, $M$, and $T$. Compute $B -M + T$.
2010 Princeton University Math Competition, 6
Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$. Find $f(100)$.
2009 AMC 8, 5
A sequence of numbers starts with $ 1$, $ 2$, and $ 3$. The fourth number of the sequence is the sum of the previous three numbers in the sequence: $ 1\plus{}2\plus{}3\equal{}6$. In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
$ \textbf{(A)}\ 11 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 37 \qquad
\textbf{(D)}\ 68 \qquad
\textbf{(E)}\ 99$
2004 Alexandru Myller, 4
Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous.
[i]Mihai Piticari[/i]
2003 All-Russian Olympiad Regional Round, 11.5
Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.
2002 Swedish Mathematical Competition, 5
The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.
2020 Taiwan APMO Preliminary, P7
[$XYZ$] denotes the area of $\triangle XYZ$
We have a $\triangle ABC$,$BC=6,CA=7,AB=8$
(1)If $O$ is the circumcenter of $\triangle ABC$, find [$OBC$]:[$OCA$]:[$OAB$]
(2)If $H$ is the orthocenter of $\triangle ABC$, find [$HBC$]:[$HCA$]:[$HAB$]
1978 AMC 12/AHSME, 19
A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$, then the probability of choosing $n$ is $p$, and if $n > 50$, then the probability of choosing $n$ is $3p$. The probability that a perfect square is chosen is
$\textbf{(A) }.05\qquad\textbf{(B) }.065\qquad\textbf{(C) }.08\qquad\textbf{(D) }.09\qquad \textbf{(E) }.1$
2014 District Olympiad, 4
Let $(G,\cdot)$ be a group with no elements of order 4, and let
$f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for
all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$
for all $x\in G$.
1996 Irish Math Olympiad, 2
Show that for every positive integer $ n$, $ 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4$.