Found problems: 85335
Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.
Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.
Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$. Prove that there is a regular $n$-gon whose all vertices lie on the considered arcs
Suppose that $ 4^a \equal{} 5$, $ 5^b \equal{} 6$, $ 6^c \equal{} 7$, and $ 7^d \equal{} 8$. What is $ a\cdot b\cdot c\cdot d$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ \frac{3}{2}\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ \frac{5}{2}\qquad
\textbf{(E)}\ 3$
Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?
$ \textbf{(A)}21\qquad\textbf{(B)}25\qquad\textbf{(C)}27\qquad\textbf{(D)}35\qquad\textbf{(E)}56 $
Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds:
$$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$ where $p$ represents the semi-perimeter of the triangle.
Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$.
For all such $a$, what is $x$?
Let $ f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $ N_k$ of natural numbers such that \[ f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\
.\] Prove that $ f$ is identically $ 1$.
[i]I. Katai[/i]
An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder?
$\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)
Some positive integer has an even number of divisors. Anya wants to split these divisors into pairs so that the products of the numbers in each pair have the same number of divisors. Prove that she can do this in exactly one way.
[i]Proposed by Oleksii Masalitin[/i]
Let $\triangle ABC$ be acute. The feet of altitudes from the corners $A, B, C$ are $ D, E, F$, respectively. If $|DF|=3, |FE|=4,$ and $|DE|=5$, then what is the radius of the circle with center $C$ and tangent to $DE$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$
Prove that
$$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$.
a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd.
b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$.
c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.
Find all positive integers $n$ and prime numbers $p$ such that $$17^n \cdot 2^{n^2} - p =(2^{n^2+3}+2^{n^2}-1) \cdot n^2.$$
[i]Authored by Nikola Velov[/i]
An $n\times n$ Latin square is a $n\times n$ grid that is filled with $n$ $1$'s, $n$ $2$'s, $\dots$, and $n$ $n$'s such that each column and row of the grid contains exactly one of each $1$, $2$, $\dots$, $n$. For example, the following is a valid $2\times2$ Latin square: $\textstyle\begin{bmatrix}2&1\\1&2\end{bmatrix}$, but this is not: $\textstyle\begin{bmatrix}2&1\\2&1\end{bmatrix}$. How many $4\times4$ Latin squares are there?
In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$, respectively. If $\angle BAC = 40^o$, what is the value of $\angle QPR$ in degrees ?
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad
\textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad
\textbf{(D) }\dfrac{2014}{2013}\qquad
\textbf{(E) }2014^{\frac1{2014}}\qquad$
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
Bokos tribus have $2021$ closed chests, we know that every chest have some amount of rupias and some amount of diamonts. They are going to do a deal with Link, that consits that Link will stay with a amount of chests and Bokos with the rest. Before opening the chests, Link has to say the amount of chest that he will stay with. After this the chests open and Link has to choose the chests with the amount that he previously said. Link doesn´t want to make Bokos angry so he wants to say the smallest number of chest that he will stay with, but guaranteeing that he stay with at least with the half of diamonts, and at least the half of the rupias. What number does Link needs to say?