Found problems: 85335
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$
Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.
Call a positive rational number in simplest terms [i]coddly[/i] if its numerator and denominator are both odd. Consider the equation $$2017= x_1\text{ }\square\text{ }x_2\text{ }\square\text{ }x_3\text{ }\ldots \text{ }\square \text{ }x_{2016} \text{ }\square \text{ }x_{2017},$$ where there are $2016$ boxes. We fill up the boxes randomly with the operations $+$, $-$, and $\times$. Compute the probability that there exists a solution in [b]distinct[/b] coddly numbers $(x_1,x_2, \ldots x_{2017})$ to the resulting equation.
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$.
(a) Prove that $P(x) > -4$ for all real $x$.
(b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.
Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$?
There are $100$ positive numbers $a_1$, $a_2$, $...$, $a_{100}$ such that $$\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.$$ Prove that $$a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.$$
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$
[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$
A ring $R$ (not necessarily commutative) contains at least one non-zero zero divisor and the number of zero divisors is finite. Prove that $R$ is finite.
A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$. What is the last page to be read?
Prove that $\frac{1}{2} \frac{3}{4} \frac{5}{6} \frac{7}{8} ... \frac{99}{100 } <\frac{1}{10}$.
Determine if there is any triple of nonnegative integers, not necessarily different, $(a, b, c)$ such that:
$$a^3 + b^3 + c^3 = 2016$$
Let $\mathbb{P}$ be the set of all primes and let $M$ be a subset of $\mathbb{P}$ with at least three elements.
Suppose that for all $k \geq 1$ and for all subsets $A=\{p_1,p_2,\dots ,p_k \}$ of $M$ ,$A\neq M$ , all prime factors of $p_1p_2\dots p_k-1$ are in $M$ . Prove that $M=\mathbb{P}$.
For a positive integer $n$, define a function $ f_n (x) $ at an interval $ [ 0, n+1 ] $ as
\[ f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . \]
Let $ a_n $ be the minimum value of $f_n (x) $. Find the value of
\[ \sum_{n=1}^{11} (-1)^{n+1} a_n . \]
A regular hexagon is placed on top of a unit circle such that one vertex coincides with the center of the circle, exactly two vertices lie on the circumference of the circle, and exactly one vertex lies outside of the circle. Determine the area of the hexagon.
Let $k_1,k_2,\ldots,k_5$ be five circles in the lane such that $k_1$ and $k_2$ are externally tangent to each other at point $T,$ $k_3$ and $k_4$ are exetrnally tangent to both $k_1$ and $k_2,$ $k_5$ is externally tangent to $k_3$ and $k_4$ at points $U$ and $V,$ respectively, and $k_5$ intersects $k_1$ at $P$ and $Q,$ like shown in the figure. Prove that \[\frac{PU}{QU}\cdot\frac{PV}{QV}=\frac{PT^2}{QT^2}.\]
Let $ f$ be a function defined on $ \text{N}_0 \equal{} \{ 0,1,2,3,...\}$ and with values in $ \text{N}_0$, such that for $ n,m \in \text{N}_0$ and $ m \leq 9, f(10n \plus{} m) \equal{} f(n) \plus{} 11m$ and $ f(0) \equal{} 0.$ How many solutions are there to the equation $ f(x) \equal{} 1995$?
A. None
B. 1
C. 2
D. 11
E. Infinitely many
Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ?
(S Zlobin)
Determine the smallest integer $n$ for which there exist integers $x_1,\ldots,x_n$ and positive integers $a_1,\ldots,a_n$ so that
\begin{align*}
x_1+\cdots+x_n &=0,\\
a_1x_1+\cdots+a_nx_n&>0, \text{ and }\\
a_1^2x_1+\cdots+a_n^2x_n &<0.
\end{align*}
Determine all positive integers $ k,n $ for which $ 2^k+10n^2+n^4 $ is a perfect square.
[i]Japan EGMO 2016 Shortlist[/i]
Find all triangles $ABC$ such that $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$, where, as usual, $a, b, c$ are the lengths of sides $BC, CA, AB$ and $R$ is the circumradius.
Find all real numbers $x$ such that $$x^3 = \{(x + 1)^3\}$$ where $\{y\}$ denotes the fractional part of $y$, i.e. the difference between $y$ and the largest integer less than or equal to $y$.
Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=2^n$ . Find the maximum possible value of positive integer $n$ .
Let $S$ be a subset of $\{1,2,3,4,5,6,7,8,9,10\}$. If $S$ has the property that the sums of three elements of $S$ are all different, find the maximum number of elements of $S$.
There are three kinds of things, which are designated respectively by the words (stripped of all common meaning) [i]notes[/i], [i]staves [/i], and [i]heads[/i]. There can be a certain relationship between a note and a head, which is expressed by the saying: they match. Also, a note and a head can match and two different staves can match. Given are the following axioms:
(a) If a note and a head each match the same stave, then they match,
(b) If two different notes both match with stave B, and also both match with head V, then B and V match,
(c) If two staves match, then there is a note that matches both,
(d) If a note and a stave are given, then there is a head that matches both.
Prove the following theorem, denoting the axiom you apply by its letter.
If three staves that differ from each other, each one matches every other, and no note matches any of the three staves, then there is a head that matches all three staves.
[hide=original wording] Er zijn drie soorten van dingen, die respectievelijk worden aangeduid met de (van alle gangbare betekenis ontdane) woorden noten, balken en vellen.
Tussen een noot en een vel kan een zekere betrekking bestaan die uitgedrukt wordt door de zegswijze: zij passen bij elkaar. Ook kunnen een noot en een vel bij elkaar passen en twee verschillende balken kunnen bij elkaar passen.
Gegeven zijn de volgende axioma’s:
(a) Als een noot en een vel elk passen bij de zelfde balk, dan passen zij bij elkaar;
(b) Als tw’ee verschillende noten beide passen bij balk b, en ook passen bij het vel v, dan passen b en v bij elkaar;
(c) Als twee balken bij elkaar passen, dan is er een noot die bij beiden past;
(d) Als een noot en een balk zijn gegeven, dan is er een vel dat bij beiden past.
Bewijs de volgende stelling en geef daarbij telkens door zijn letter het axioma aan dat U toepast.
Als van drie onderling verschillende balken elke past bij elke andere en er geen noot bij de drie balken past, dan is er een vel dat bij alle drie de balken past.[/hide]