Found problems: 85335
Let $A$ be a set of $8$ elements, and $B := (B_1,...,B_7)$ be an ordered $7$-tuple of subsets of $A$. Let $N$ be the number of such $7$-tuples $B$ such that there exists a unique $4$-element subset $I \subseteq \{1,2,...,7\}$ for which the intersection $\cap _{ i\in I} B_i$ is nonempty. Find the remainder when $N$ is divided by $67$.
Let $f(N) = N \left( \frac{9}{10} \right)^N$ , and let $\frac{m}{n}$ denote the maximum value of $f(N)$, as $N$ ranges over the positive integers. If $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1000$.
Given two circles $k_1$ and $k_2$ which intersect at $Q$ and $Q'.$ Let $P$ be a point on $k_2$ and inside of $k_1 $ such that the line $PQ$ intersects $k_1$ in a point $X\ne Q$ and such that the tangent to $k_1$ at $X$ intersects $k_2$ in points $A$ and $B.$ Let $k$ be the circle through $A,B$ which is tangent to the line through $P$ parallel to $AB.$
Prove that the circles $k_1$ and $k$ are tangent.
In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$?
Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers.
(The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)
Find the least positive integer $n$ such that $2^{2014}$ divides $19^n-1$.
Find all triplets of positive integers $(a,b,c)$ for which the number $3^a+3^b+3^c$ is a perfect square.
Let $f(X)$ be a monic irreducible polynomial over $\mathbb{Z}$; therefore, by Gauss's Lemma, $f$ is also irreducible over $\mathbb{Q}$ (you may assume this). Moreover, assume $f(X) \mid f\left(X^2+n\right)$ where $n$ is an integer such that $n \notin\{-1,0,1\}$. Show that $n^2 \nmid f(0)$.
Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds:
$$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$
[i]D. Zmiaikou[/i]
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
Determine all real numbers $a,b,c,d$ that satisfy the following equations
\[\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}\]
Alma places spies on some of the squares on a $2020\times 2020$ game board. Now Bertha secretly chooses a quadradic area consisting of $1020 \times 1020$ squares and tells Alma which spies are standing on a square in the secret quadradic area. At least how many spies must Alma have placed in order for her to determine with certainty which area Bertha has chosen?
Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $
[b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $
[b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.
Solve in integers the system of equations: $$x^2-y^2=z$$
$$3xy+(x-y)z=z^2$$
Let $a,b$ be integers and $p$ be a prime number such that:
(i) $p$ is the greatest common divisor of $a$ and $b$;
(ii) $p^2$ divides $a$.
Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.
Is it possible to place a triangle with area $1999$ and perimeter $19992$ in the interior of a triangle with area $2000$ and perimeter $20002$?
A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for $1$ teaching assistant to grade all the homework assignments.
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
Sketch the graph of $u(x, 1)$, if $0 \le x\le1$,
\[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]
Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number?
$\textbf{(A)}~\frac12 \qquad
\textbf{(B)}~\frac23 \qquad
\textbf{(C)}~\frac34 \qquad
\textbf{(D)}~\frac56\qquad
\textbf{(E)}~\frac78$
Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?
Natural $a,b,c$ are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every $a \times a$, every $b \times b$, every $c \times c$ squares is even.
Is it true, that every number in table must be even?
[color=darkred]Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions
\[f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}\]
such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$[/color]
Let $ABCDEF$ be the regular hexagon. It is known that the area of the triangle $ACD$ is equal to $8$. Find the hexagonal area of $ABCDEF$.