Found problems: 85335
All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$
Let the inequality $$Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$$
with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$. Is it possible to construct a triangle with sides of lengths $a, b, c$?
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
Let $f$ be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that $f$ satisfies the following conditions:
[b](1)[/b] $f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);$
[b](2)[/b] $f(a,1-a)=1$
Prove that $f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1$.
$a)$ An equilateral triangle of sides $a$ is given and a triangle $MNP$ is constructed under the following conditions: $P\in (AB),M\in (BC),N\in (AC)$, such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$. Find the length of the segment $MP$.
$b)$ Show that for any acute triangle $ABC$ one can find points $P\in (AB),M\in (BC),N\in (AC)$ such that $MP\perp AB,NM\perp BC$ and $PN\perp AC$.
An [i]anti-Pascal[/i] triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$.
\[\begin{array}{
c@{\hspace{4pt}}c@{\hspace{4pt}}
c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c
} \vspace{4pt}
& & & 4 & & & \\\vspace{4pt}
& & 2 & & 6 & & \\\vspace{4pt}
& 5 & & 7 & & 1 & \\\vspace{4pt}
8 & & 3 & & 10 & & 9 \\\vspace{4pt}
\end{array}\]
Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?
[i]Proposed by Morteza Saghafian, Iran[/i]
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits?
[i]Proposed by A. Shapovalov[/i]
Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all possible values for the first integer.
The non-negative real numbers $x, y, z$ satisfy the equality $x + y + z = 1$.
Determine the highest possible value of the expression $E (x, y, z) = (x + 2y + 3z) (6x +3y + 2z)$.
Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?
Compute $1+2\cdot3^4$.
[i]Proposed by Evan Chen[/i]
Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length.
1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints.
2. Find all possible numbers of the diagonals with two black endpoints.
Given natural numbers $n > k > 1$, find all real solutions $x_1,..., x_n$ of the system
$$x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2$$ for 1 $\le i \le n$. Here $x_{n+i} = x_i$ for all$ i$.
Points $M$ and $M'$ are isogonal conjugates in the traingle $ABC$. We draw perpendiculars $MP$, $MQ$, $MR$, and $M'P'$, $M'Q'$, $M'R'$ to the sides $BC$, $AC$, $AB$ respectively. Let $QR$, $Q'R'$, and $RP$, $R'P'$ and $PQ$, $P'Q'$ intersect at $E$, $F$, $G$ respectively. Show that the lines $EA$, $FB$, and $GC$ are parallel.
The Fibonacci numbers $F_n$ are defined as $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n> 2$. Let $A$ be the minimum area of a (possibly degenerate) convex polygon with $2020$ sides, whose side lengths are the first $2020$ Fibonacci numbers $F_1$, $F_2$, $...$ , $F_{2020}$ (in any order). A degenerate convex polygon is a polygon where all angles are $\le 180^o$. If $A$ can be expressed in the form $$\frac{\sqrt{(F_a-b)^2-c}}{d}$$ , where $a, b, c$ and $d$ are positive integers, compute the minimal possible value of $a + b + c + d$.
Let the incircle of triangle $ ABC$ touch sides $ BC,\, CA$ and $ AB$ at $ D,\, E$ and $ F,$ respectively. Let $ \omega,\,\omega_{1},\,\omega_{2}$ and $ \omega_{3}$ denote the circumcircles of triangle $ ABC,\, AEF,\, BDF$ and $ CDE$ respectively.
Let $ \omega$ and $ \omega_{1}$ intersect at $ A$ and $ P,\,\omega$ and $ \omega_{2}$ intersect at $ B$ and $ Q,\,\omega$ and $ \omega_{3}$ intersect at $ C$ and $ R.$
$ a.$ Prove that $ \omega_{1},\,\omega_{2}$ and $ \omega_{3}$ intersect in a common point.
$ b.$ Show that $ PD,\, QE$ and $ RF$ are concurrent.
A positive integer $b\geq 2$ is [i]neat[/i] if and only if there exist positive base-$b$ digits $x$ and $y$ (that is, $x$ and $y$ are integers, $0<x<b$ and $0<y<b$) such that the number $x.y$ base $b$ (that is, $x+\tfrac yb$) is an integer multiple of $x/y$. Find the number of [i]neat[/i] integers less than or equal to $100$.
A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$
\[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \]
Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$
Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$
Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.
Let $S$ be a set of nonnegative integers such that
[list]
[*] there exist two elements $a$ and $b$ in $S$ such that $a,b>1$ and $\gcd(a,b)=1$; and
[*] for any (not necessarily distinct) element $x$ and nonzero element $y$ in $S$, both $xy$ and the remainder when $x$ is divided by $y$ are in $S$.
[/list]
Prove that $S$ contains every nonnegative integer.
[i]Jacob Paltrowitz[/i]
The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$.