Found problems: 85335
George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.
[i]Proposed by Carl Schildkraut and Holden Mui[/i]
The diagram below shows two concentric circles whose areas are $7$ and $53$ and a pair of perpendicular lines where one line contains diameters of both circles and the other is tangent to the smaller circle. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/3/b/87cbb97a799686cf5dbec9dcd79b6b03e1995c.png[/img]
Methanol can be gently oxidized with hot copper metal. What is(are) the product(s) of this oxidation?
$ \textbf{(A) }\text{Acetic acid}\qquad\textbf{(B) }\text{Carbon dioxide + Water}\qquad\textbf{(C) }\text{Ethanol} \qquad\textbf{(D) }\text{Methanal} \qquad$
Is there a real number $\alpha$ such that $\cos\alpha$ is irrational but $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are all rational? (Author: V. Senderov)
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$
Given two positive integers $n$ and $k$, we say that $k$ is [i]$n$-ergetic[/i] if:
However the elements of $M=\{1,2,\ldots, k\}$ are coloured in red and green, there exist $n$ not necessarily distinct integers of the same colour whose sum is again an element of $M$ of the same colour. For each positive integer $n$, determine the least $n$-ergetic integer, if it exists.
For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$
There are $11$ empty boxes. In one move you can put one coin in some 10 of them. Two people play and take turns. Wins the one after which for the first time there will be $21$ coins in one of the boxes. Who wins when played correctly?
Let $ \varepsilon_1,\varepsilon_2,...,\varepsilon_{2n}$ be independent random variables such that $ P(\varepsilon_i\equal{}1)\equal{}P(\varepsilon_i\equal{}\minus{}1)\equal{}\frac 12$ for all $ i$, and define $ S_k\equal{}\sum_{i\equal{}1}^k \varepsilon_i, \;1\leq k \leq 2n$. Let $ N_{2n}$ denote the number of integers $ k\in [2,2n]$ such that either $ S_k>0$, or $ S_k\equal{}0$ and $ S_{k\minus{}1}>0$. Compute the variance of $ N_{2n}$.
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]
A [i]chaisri[/i] figure is a triangle which the three vertices are vertices of a regular $2019$-gon. Two different chaisri figure may be formed by different regular $2019$-gon.
A [i]thubkaew[/i] figure is a convex polygon which can be dissected into multiple chaisri figure where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.
Determine the maximum number of vertices that a thubkaew figure may have.
In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$.
(Grigory Filippovsky)
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape?
[i]( Proposed by Vlad Spǎtaru, Romania)[/i]
Let $n$ be a positive integer. There are $n$ colors available. Each of the integers from $1$ to $1000$ must be painted with one of the $n$ colors such that any two different numbers, if one divides the other, are painted in different colors. Determine the smallest value of $n$ for which this is possible.
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$
In quadrilateral $ABCD,$ suppose that $\overline{CD}$ is perpendicular to $\overline{BC}$ and $\overline{DA}$. Point $E$ is chosen on segment $\overline{CD}$ such that $\angle AED = \angle BEC$. If $AB = 6$, $AD = 7$, and $\angle ABC = 120^o$ , compute $AE + EB$.
For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $$x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$$
Prove that for every $n$, it also satisfies $$x_1+x_2/2+x_3/3+...+x_n/n\le 3$$
Let $ABC$ be an non-isosceles triangle with incenter $I$, circumcenter $O$ and a point $D$ on segment $BC $such that $(BID) $cut segments $AB $ at$ E $and $(CID) $cuts segment $AC $at $F$ Circle $(DEF)$ cuts segments $AB$,$AC $again at $M,N$. Let $P$ The intersection of $IB$ and $DE $ , $Q$ The intersection of $IC$and $DF$ . Prove that $EN,FM,PQ $are parallel and the median of vertex $I$in triangle $IPQ$ bisects the arc $BAC$ of $(O)$.
In the triangle $ABC$ the angle bisectors $AD$ and $BE$ are drawn. Prove that $\angle ACB = 60 {} ^ \circ$ if and only if $AE + BD = AB$.
(Hilko Danilo)
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ lying in the interior. Let $E$ and $F$ be the midpoints of the segments $BC$ and $AD$, respectively. Let $X$ be the point lying on the same side of the line $EF$ as the vertex $C$ such that $\triangle EXF$ and $\triangle BOA$ are similar. Prove that $XC = XD$.
Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$
A salmon in a mountain river must overpass two waterfalls. In every minute, the probability of the salmon to overpass the first waterfall is $p > 0$, and the probability to overpass the second waterfall is $q > 0$. These two events are assumed to be independent. Compute the probability that the salmon did not overpass the first waterfall in $n$ minutes, assuming that it did not overpass both waterfalls in that time.
You are responsible for arranging a banquet for an agency. In the agency, some pairs of agents are enemies. A group of agents are called [i]avengers[/i], if and only if the number of agents in the group is odd and at least $3$, and it is possible to arrange all of them around a round table so that every two neighbors are enemies.
You figure out a way to assign all agents to $11$ tables so that any two agents on the same tables are not enemies, and that’s the minimum number of tables you can get. Prove that there are at least $2^{10}-11$ avengers in the agency.
This problem is adapted from 2015 IMO Shortlist C7.