Found problems: 85335
LMT Accuracy Rounds, 2022 S5
A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?
1989 Nordic, 3
Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.
1980 IMO, 11
A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations
\[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\]
Prove that $(ABC)$ is equilateral and $D$ is its center.
2022 Taiwan TST Round 1, N
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
1983 Tournament Of Towns, (043) A5
$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied:
If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$.
Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation.
(M Kontsevich, Moscow)
2020 Online Math Open Problems, 30
Let $c$ be the smallest positive real number such that for all positive integers $n$ and all positive real numbers $x_1$, $\ldots$, $x_n$, the inequality \[ \sum_{k=0}^n \frac{(n^3+k^3-k^2n)^{3/2}}{\sqrt{x_1^2+\dots +x_k^2+x_{k+1}+\dots +x_n}} \leq \sqrt{3}\left(\sum_{i=1}^n \frac{i^3(4n-3i+100)}{x_i}\right)+cn^5+100n^4 \] holds. Compute $\lfloor 2020c \rfloor$.
[i]Proposed by Luke Robitaille[/i]
2012 Vietnam National Olympiad, 2
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
2015 Harvard-MIT Mathematics Tournament, 3
Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.
2022 Portugal MO, 4
Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.
2015 ASDAN Math Tournament, 1
Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.
2005 Grigore Moisil Urziceni, 1
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system:
$$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$
2014 AIME Problems, 11
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2017 Junior Balkan Team Selection Tests - Romania, 1
If $a, b, c \in [-1, 1]$ satisfy $a + b + c + abc = 0$, prove that $a^2 + b^2 + c^2 \ge 3(a + b + c)$ .
When does the equality hold?
PEN A Problems, 72
Determine all pairs $(n,p)$ of nonnegative integers such that [list] [*] $p$ is a prime, [*] $n<2p$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]
2024 Mathematical Talent Reward Programme, 2
Find positive reals $a,b,c$ such that: $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} = 2$$
2015 District Olympiad, 2
[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even.
[b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural.
2014 HMNT, 8
Let $a, b, c, x$ be reals with $(a + b)(b + c)(c + a) \ne 0$ that satisfy
$$\frac{a^2}{a + b}=\frac{a^2}{a + c}+ 20, \,\,\, \frac{b^2}{b + c}=\frac{b^2}{b + a}+ 14, \text{and}\,\,\, \frac{c^2}{c + a}=\frac{c^2}{c + b}+ x.$$
Compute $x$.
2012 Today's Calculation Of Integral, 771
(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.
(2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.
Brazil L2 Finals (OBM) - geometry, 2014.2
Let $AB$ be a diameter of the circunference $\omega$, let $C$ and $D$ be point in this circunference, such that $CD$ is perpedicular to $AB$. Let $E$ be the point of intersection of the segment $CD$ and the segment $AB$, and a point $P$ that is in the segment $CD, P$ is different of $E$. The lines $AP$ and $BP$ intersects $\omega$, in $F$ and $G$ respectively. If $O$ is the circumcenter of triangle $EFG$, show that the area of triangle $OCD$ is invariant, independent of the position of the point $P$.
2019 OMMock - Mexico National Olympiad Mock Exam, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$
[i]Proposed by Victor DomÃnguez and Pablo Valeriano[/i]
2014 Peru IMO TST, 9
Prove that for every positive integer $n$ there exist integers $a$ and $b,$ both greater than $1,$ such that $a ^ 2 + 1 = 2b ^ 2$ and $a - b$ is a multiple of $n.$
2013 China Team Selection Test, 1
Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$.
Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$
2017 F = ma, 10
10) The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the bottom of the jug? $\rho$ is the density of the milk.
A) $P = \rho gh$
B) $P = \rho gH$
C) $\rho gH< P < \rho gh$
D) $P > \rho gh$
E) $P < \rho gH$
2022 IMO Shortlist, A7
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
2010 ISI B.Stat Entrance Exam, 4
A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.