Found problems: 85335
For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that
\[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\]
holds.
For any positive integer $m$, denote by $d_i(m)$ the number of positive divisors of $m$ that are congruent to $i$ modulo $2$. Prove that if $n$ is a positive integer, then
\[\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.\]
Let acute scalene $\Delta ABC$ have circumcircle $\omega$. Let $M$ be the midpoint of $BC$, define $X$ as the other intersection of $AM$ with $\omega$. Let $E,F$ be the feet of altitudes from $B,C$ to $AC, AB$ respectively. Let $Q$ be the second intersection of the circumcircle of $\Delta AEF$ and $\omega$. Let $Y\neq X$ be a point such that $MX=MY$ and $QMXY$ is cyclic. Finally, let $S$ be a point on $BC$ such that $\angle BAS=\angle MAC.$ Prove that the quadrilaterals $BFYS$ and $CEYS$ are cyclic.
Proposed by Kanav Talwar and Malay Mahajan
Real numbers $a$, $b$, $c$, and $d$ satisfy the system of equations
\begin{align*}
-a-27b-8d &= 1, \\
8a+64b+c+27d &= 0, \\
27a+125b+8c+64d &= 1, \\
64a+216b+27c+125d &= 8.
\end{align*}
Find $12a+108b+48d$.
[i]Proposed by firebolt360[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions:
a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$
b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval.
[i]Proposed by Navid Safaei[/i]
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$.
(b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.
A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?
[asy]
path p=origin--(1,0)--(1,1)--(0,1)--cycle;
draw(p);
draw(shift(3,0)*p);
draw(shift(3,1)*p);
draw(shift(4,0)*p);
draw(shift(4,1)*p);
draw(shift(7,0)*p);
draw(shift(7,1)*p);
draw(shift(7,2)*p);
draw(shift(8,0)*p);
draw(shift(8,1)*p);
draw(shift(8,2)*p);
draw(shift(9,0)*p);
draw(shift(9,1)*p);
draw(shift(9,2)*p);
[/asy]
$ \text{(A)}\ 11\qquad\text{(B)}\ 12\qquad\text{(C)}\ 13\qquad\text{(D)}\ 14\qquad\text{(E)}\ 15 $
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
Prove that if $x, y$ and $n$ are positive integers such that $$x^{2024} + y^{2024} = 2^n,$$ then $x=y$.
On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?
In a right triangle, the length of one side is a prime and the lengths of the other
side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.
In any triangle $ABC$ prove that the following relationship holds:
$$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$
[i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]
A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?
Let $n$ be positive integers and $t$ be a positive real number.
Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$
The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered?
[i]Proposed by Evan Chang (squareman), USA[/i]
Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.
\[\int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac{\arctan(x)\log^2(x)}{x}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
Calculate with at most $10\%$ relative error
\[\int_{-\infty}^{\infty}(x^4+4x+4)^{-100}dx\]
The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$.
[i]A. Kuznetsov[/i]
Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Find the maximum possible value of $x^2+xy+2y^2$.
A digital calendar displays the date: day, month, and year, with $2$ digits for the day, $2$ digits for the month, and $2$ digits for the year. For example, $01-01-01$ is January $1$, $2001$ and $05-25-23$ is May $25$, $2023$. In front of the calendar is a mirror. The digits of the calendar are as in the figure
[img]https://cdn.artofproblemsolving.com/attachments/c/5/a08a4e34071fff4d33b95b23690254f55b33e1.gif[/img]
If $0, 1, 2, 5$, and $8$ are reflected, respectively, in $0, 1, 5, 2$, and $8$, and the other digits lose meaning when reflected, determine how many days of the century, when reflected in the mirror, also correspond to a date.
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.
Let $ a $ and $ b $ be different real numbers. Prove that for any real numbers $ c_1, c_2, \ldots,c_n $ there exists a sequence of $ n $-elements $ (x_i) $, each term of which is equal to one of the numbers $ a $ or $ b $ such that $$
|x_1c_1 + x_2c_2 + \ldots + x_nc_n| \geq \frac{|b-a|}{2}(|c_1|+|c_2|+\ldots+|c_n|).$$
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
The fraction $ \frac{1}{3}$:
$ \textbf{(A)}\ \text{equals 0.33333333} \qquad
\textbf{(B)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\
\textbf{(C)}\ \text{is less than 0.33333333 by }\frac{1}{3 \cdot 10^9} \\
\textbf{(D)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^8} \\
\textbf{(E)}\ \text{is greater than 0.33333333 by }\frac{1}{3 \cdot 10^9}$