Found problems: 85335
For a positive integer $n$, define the $n$th triangular number $T_n$ to be $\frac{n(n+1)}{2}$, and define the $n$th square number $S_n$ to be $n^2$. Find the value of \[\sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}.\]
[i]Proposed by Yannick Yao[/i]
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that
\[2^{j-i}x_ix_j>2^{s-3}.\]
The racing tournament has $12$ stages and $ n $ participants. After each stage, all participants, depending on the occupied place $ k $, receive points $ a_k $ (the numbers $ a_k $ are natural and $ a_1> a_2> \dots> a_n $). For what is the smallest $ n $ the tournament organizer can choose the numbers $ a_1 $, $ \dots $, $ a_n $ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.
4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.
Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]
Somewhere in the universe, $n$ students are taking a $10$-question math competition. Their collective performance is called [i]laughable[/i] if, for some pair of questions, there exist $57$ students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest $n$ such that the performance is necessarily laughable.
Let $ x$, $ y$, $ z$ be positive numbers. Find the minimum value of:
$ (a)\quad \frac{x^2 \plus{} y^2 \plus{} z^2}{xy \plus{} yz}$
$ (b)\quad \frac{x^2 \plus{} y^2 \plus{} 2z^2}{xy \plus{} yz}$
We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$.
Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible?
[i]Proposed by Krit Boonsiriseth[/i]
A sequence $a_1,a_2,a_3,\ldots $ of non-negative integers is such that $a_{n+1}$ is the last digit of $a_n^n+a_{n-1}$ for all $n>2$. Is it always true that for some $n_0$ the sequence $a_{n_0},a_{n_0+1},a_{n_0+2},\ldots$ is periodic?
For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$.
They are defined inductively, by the following recurrences.
$A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$
$B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$
Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.
Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square.
Given an integer \( n \geq 1 \), Jo-Ané alternately writes crosses (\( \mathcal{X} \)) and circles (\( \mathcal{O}\)) in the cells of a square grid with \( 2n + 1 \) rows and \( 2n + 1 \) columns: she first writes a cross in a cell, then a circle in a second cell, then a cross in a third cell, and so on. When the table is completely filled, her score is calculated as the sum \( \mathcal{X}+ \mathcal{O} \), where \( \mathcal{X} \) is the number of rows containing more crosses than circles and \( \mathcal{O} \) is the number of columns containing more circles than crosses.
Determine, in terms of \( n \), the highest possible score that Jo-Ané can obtain..
Prove that for any positive real numbers $a_i,b_i,c_i$ with $i=1,2,3$,
$(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)$
Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
[i]Proposed by Ankan Bhattacharya[/i]
Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that:
[b]a)[/b] $ \text{tr} (A) =\det (A) =0 $
[b]b)[/b] $ AB^nA=0, $ for any natural number $ n $
Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.
Let $A$ be the set of permutations $a=(a_1,a_2,…,a_n)$ of $M=\{1,2,…n\}$ with the following property: There doesn’t exist a subset $S$ of $M$ such that $a(S)=S$. For $\forall$ such permutation $a$ let $d(a)=\sum_{k=1}^n (a_k-k)^2$ . Determine the smallest value of $d(a)$.
Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.
Find all values of $b$ such that the difference between the maximum and minimum values of $f(x) = x^2-2bx-1$ on the interval $[0, 1]$ is $1$.
The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule:
\[a_{n+1}=a_{n}+2t(n)\]
for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?
If $ A \star B$ means $ \frac{A\plus{}B}{2}$, then $ (3 \star 5) \star 8$ is
\[ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 30
\]
Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.
The vertices of $\Delta ABC$ lie on the graphics of the function $f(x)=x^2$ and its centroid is $M(1,7)$. Determine the greatest possible value of the area of $\Delta ABC$.
Let $f:[0,1]\to(0,1)$ be a surjective function.
[list=a]
[*]Prove that $f$ has at least one point of discontinuity.
[*]Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity.
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]