Found problems: 85335
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.
We have a triangle with $ \angle BAC=\angle BCA. $ The point $ E $ is on the interior bisector of $ \angle ABC $ so that $ \angle EAB =\angle ACB. $ Let $ D $ be a point on $ BC $ such that $ B $ is on the segment $ CD $ (endpoints excluded) and $ BD=AB. $ Show that the midpoint of $ AC $ is on the line $ DE. $
Find the sum of the indexes of the singular points other than zero of the vector field
\[z\overline{z}^2+z^4+2\overline{z}^4\]
If $0<a<b$ then $$\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0$$
Calculate the following indefinite integrals.
[1] $\int (2x+1)\sqrt{x+2}\ dx$
[2] $\int \frac{1+\cos x}{x+\sin x}\ dx$
[3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$
[4] $\int \frac{(x-3)^2}{x^4}\ dx$
[5] $\int \frac{dx}{\tan x}\ dx$
Let $a,b$ be positive real numbers satisfying $2ab=a-b$. Denote for any positive integer $k$ $x_k$ and $y_k$ to be the closest integer to $ak$ and $bk$, respectively (if there are two closest integers, choose the larger one). Prove that any positive integer $n$ appears in the sequence $(x_k)_{k\ge 1}$ if and only if it appears at least three times in the sequence $(y_k)_{k\ge 1}$.
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$.
(A. Voidelevich)
Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell [i]correct [/i] if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be [i]correct[/i]?
Given is an acute triangle \( \triangle ABC \) with \( AB < AC \). Let \( M \) be the midpoint of side \( BC \), and let \( X \) and \( Y \) be points on segments \( BM \) and \( CM \), respectively, such that \( BX = CY \). Let \( \omega_1 \) be the circumcircle of \( \triangle ABX \), and \( \omega_2 \) the circumcircle of \( \triangle ACY \). The common tangent \( t \) to \( \omega_1 \) and \( \omega_2 \), which lies closer to point \( A \), touches \( \omega_1 \) and \( \omega_2 \) at points \( P \) and \( Q \), respectively. Let the line \( MP \) intersect \( \omega_1 \) again at \( U \), and the line \( MQ \) intersect \( \omega_2 \) again at \( V \). Prove that the circumcircle of triangle \( \triangle MUV \) is tangent to both \( \omega_1 \) and \( \omega_2 \).
Let $ABC$ be an acute-angled triangle. The tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ respectively meet at $D$. The circumcircles of triangles $ABD$ and $ACD$ meet line $BC$ at additional points $E$ and $F$ respectively. Lines $DB$ and $DC$ meet the circumcircle of triangle $DEF$ at additional points $X$ and $Y$ respectively. Let $O$ be the circumcentre of triangle $DEF$. Prove that the circumcircles of triangles $ABC$ and $OXY$ are tangent to each other.
Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $
[b]a)[/b] For which integer values of $ k $ the above function is injective?
[b]b)[/b] For which integer values of $ k $ the above function is surjective?
[b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions:
$$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$
$$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$
[i]Cristinel Mortici[/i]
Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.
How many pairs of integers $(x,y)$ are there such that $2x+5y=xy-1$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 12
$
A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number
greater or equal to $x$, and no two of these cells are on the same column or row.
[i]Proposed by CSJL.[/i]
How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
Let $S = \{1, 2, 3, 5, 6, 10, 15, 30\}$. For each of the $64$ ordered pairs $(a, b)$ of elements of $S$, AJ computes $gcd(a, b)$. They then sum all of the numbers they computed. What is AJ’s sum?
Prove that there does not exist an integer $n>1$ such that $n$ divides $3^n-2^n$.
$$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\sin(2x)\,dx$$
[i]Proposed by Connor Gordon[/i]
Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle.
by A.Khachaturyan
Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent.
Maxim Didin
In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an [i]antitour[/i] from $A$ if
[list]
[*] every city (including $A$) appears in the list exactly once, and
[*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights.
[/list]
Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.
[i]Proposed by Sutanay Bhattacharya[/i]
Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$ . Prove that the line $XY$ is tangent to $\Omega$.
In a trapezoid $ABCD$, $AD \parallel BC$ and $\angle A = 60^o$. Let $E$ be a point on $AB$, and let $O_1$ and $O_2$ be circumcenters of $\vartriangle AED$ and $\vartriangle BEC$, respectively. Let $\frac{\overline{O_1O_2}}{\overline{DC}}$ be $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.