Found problems: 85335
Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.
Find all real solutions $(x,y)$ of the system $x^2+y=12=y^2+x$.
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
[i]Ray Li.[/i]
Let $n \geq 2$ be a positive integer and let $A, B$ be two finite sets of integers such that $|A| \leq n$. Let $C$ be a subset of the set $\{(a, b) | a \in A, b \in B\}$. Achilles writes on a board all possible distinct differences $a-b$ for $(a, b) \in C$ and suppose that their count is $d$. He writes on another board all triplets $(k, l, m)$, where $(k, l), (k, m) \in C$ and suppose that their count is $p$. Show that $np \geq d^2.$
Emily’s broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily’s clock also does not tick, but rather updates continuously.
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
Let $N$ be a positive integer. Consider a $N \times N$ array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A [i]move[/i] consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
What is the minimal number of additional cells that one has to color black such that, after a finite number of moves, a completely black board can be reached?
In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.
An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants.
Given integer $d>1,m$,prove that there exists integer $k>l>0$, such that $$(2^{2^k}+d,2^{2^l}+d)>m.$$
For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$
(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.
(b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$
Toss the coin $n$ times, assume that each time, only appear only head or tail
Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times
Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times
$(1)$ Find $a(2016)$ and $b(2016)$
$(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
There is a single coin on each square of a $5 \times 5$ board. All the coins look the same. Two of them are fakes and have equal weight. Genuine coins are heavier than fake ones and also weigh the same. The fake coins are on the squares sharing just one vertice. Is it possible to determine for sure a) 13 genuine coins; b) 15 genuine coins; and c) 17 genuine coins in a single weighing on a balance with no unit weights?
[i]Rustem Zhenodarov, Alexandr Gribalko, Sergey Tokarev[/i]
Let $ABC$ be a triangle. Let $\theta$ be a fixed angle for which \[\theta<\frac12\min(\angle A,\angle B,\angle C).\]
Points $S_A$ and $T_A$ lie on segment $BC$ such that $\angle BAS_A=\angle T_AAC=\theta$. Let $P_A$ and $Q_A$ be the feet from $B$ and $C$ to $\overline{AS_A}$ and $\overline{AT_A}$ respectively. Then $\ell_A$ is defined as the perpendicular bisector of $\overline{P_AQ_A}$.
Define $\ell_B$ and $\ell_C$ analogously by repeating this construction two more times (using the same value of $\theta$). Prove that $\ell_A$, $\ell_B$, and $\ell_C$ are concurrent or all parallel.
Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$
Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2m+1$ integers
$$ 2^{n}-m,2^{n}-(m-1),\ldots,2^{n}+(m-1),2^{n}+m$$
is positive and composite.
Let $n \geq 3$ be an integer and let $a,b\in\mathbb{R}$ such that $nb\geq a^2$. We consider the set \[ X = \left\{ (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n \mid \sum_{k=1}^n x_k = a, \ \sum_{k=1}^n x_k^2 = b \right\} . \] Find the image of the function $M: X\to \mathbb{R}$ given by \[ M(x_1,x_2,\ldots,x_n) = \max_{1\leq k\leq n} x_k . \]
[i]Dan Schwarz[/i]
There are \(n\) numbers arranged in a circle, and each number equals the absolute value of the difference between its two neighbors. Is it necessarily true that all numbers are equal to zero if:
a) \(n=2025\);
b) \(n=2024\)?
[i]Proposed by Anton Trygub[/i]
Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.
Let $x$ be a real number such that $3 \sin^4 x -2 \cos^6 x = -\frac{17}{25}$ . Then $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.
Let $n>1$ be a positive integer. Find the number of binary strings $(a_1, a_2, \ldots, a_n)$, such that the number of indices $1\leq i \leq n-1$ such that $a_i=a_{i+1}=0$ is equal to the number of indices $1 \leq i \leq n-1$, such that $a_i=a_{i+1}=1$.
Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.
Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is
(A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.