Found problems: 85335
In how many ways can $1, 2,\ldots, 2n$ be arranged in a $2 \times n$ rectangular array $\left(\begin{array}{cccc}a_1& a_2 & \cdots & a_n\\b_1& b_2 & \cdots & b_n\end{array}\right)$ for which:
[b](i)[/b] $a_1 < a_2 < \cdots < a_n,$
[b](ii) [/b] $b_1 < b_2 <\cdots < b_n,$
[b](iii) [/b]$a_1 < b_1, a_2 < b_2, \ldots, a_n < b_n \ ?$
There is a chessboard with $m$ columns and $n$ rows. In each blanks, an integer is given. If a rectangle $R$ (in this chessboard) has an integer $h$ satisfying the following two conditions, we call $R$ as a 'shelf'.
(i) All integers contained in $R$ are bigger than $h$.
(ii) All integers in blanks, which are not contained in $R$ but meet with $R$ at a vertex or a side, are not bigger than $h$.
Assume that all integers are given to make shelves as much as possible. Find the number of shelves.
Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\]
for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$
[b](i)[/b] Prove that $S$ is an infinite set.
[b](ii)[/b] Find the least positive integer in $S.$
[b](iii)[/b] If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \] show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)
Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions:
$f(1)=1$
$f(a+b+ab)=a+b+f(ab)$
Find the value of $f(2015)$
Proposed by Jose Antonio Gomez Ortega
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Prove that for the arbitrary positive $a_1, a_2, ... , a_n$ the following inequality is held
$$\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}$$
In triangle $ABC$, $\angle BAC= 60^o$. Angle bisector of $\angle ABC$ meets side $AC$ at $X$ and angle bisector of $\angle BCA$ meets side $AB$ at $Y$. Prove that if $I$ is the incenter of triangle $ABC$, then $IX=IY$.
For a class of $20$ students several field trips were arranged. In each trip at least one student participated. Prove that there was a field trip such that each student who participated in it took part in at least $1/20$-th of all field trips.
The integer $111111$ is the product of five prime numbers. Determine the sum of these prime numbers.
Let $ABC$ be an acute-angled triangle, $C'$ and $A'$ be arbitrary points on the sides $AB$ and $BC$ respectively, and $B'$ be the midpoint of the side $AC$.
(a) Prove that the area of triangle $A'B'C'$ is at most half the area of triangle $ABC$.
(b) Prove that the area of triangle $A'B'C'$ is equal to one fourth of the area of triangle $ABC$ if and only if at least one of the points $A'$, $C'$ is the midpoint of the corresponding side.
(E Cherepanov)
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidena plane. Sarah can only move $1$ unit in the positive $x$ or $y$ direction, and Rachel can only move $1$ unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move, and move to their new position. Sarah catches Rachel if Sarah and Rachel are every at the same point. Rachel wins if she is able to get $(0,0)$ without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?
Take a sequence $(a_n)_{n=1}^\infty$ such that
$a_1=3$
$a_n=a_1a_2a_3...a_{n-1}-1$
[b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence.
[b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.
Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.
Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. Show that the line $PE$ is tangent to the circle through $A$, $C$, and $E$.
Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.
Solve the equation $$ |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.$$
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.
In a plane, we are given line $\ell$, two points $A$ and $B$ neither of which lies on line $\ell$, and the reflection $A_1$ of point $A$ across line $\ell$. Using only a straightedge, construct the reflection $B_1$ of point $B$ across line $\ell$.
Prove that your construction works.
Note: “Using only a straightedge” means that you can perform only the following operations:
(a) Given two points, you can construct the line through them.
(b) Given two intersecting lines, you can construct their intersection point.
(c) You can select (mark) points in the plane that lie on or off objects already drawn in the plane. (The only facts you can use about these points are which lines they are on or not on.)
Do there exist $\{x,y\}\in\mathbb{Z}$ satisfying $(2x+1)^{3}+1=y^{4}$?
Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$