Found problems: 85335
Let $I$ be the incenter of triangle $ABC$. The circle of centre $A$ and radius $AI$ intersects the circumcircle of triangle $ABC$ in $M$ and $N$. Prove that the line $MN$ is tangent to the incircle of triangle $ABC$
Jorn wants to cheat at the role play: he intends to cheat the sides to re-label its two octahedra, so that each of the numbers from $1$ to $16$ has the same probability as the sum of the dice occurs. So that the game master does not notice this so easily, he only wants to use numbers from $0$ to $8$ , if necessary several times or not at all. Is this possible?
There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that
\[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \]
where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?
$ABCD $ is a cyclic quadrilateral such that the diagonals $AC $ and $BD $ are perpendicular and their intersection is $P $. Point $Q $ on the segment $CP$ is such that $CQ=AP $. Prove that the perimeter of triangle $BDQ $ is at least $2AC $.
Tuymaada 2017 Q2 Juniors
If $S=\{s_1,s_2,\dots,s_n\}$ is a set of integers with $s_1<s_2<\dots<s_n$, define
$$f(S)=\sum_{k=1}^n (-1)^k k^2 s_k.$$
(If $S$ is empty, $f(S)=0$.) Compute the average value of $f(S)$ as $S$ ranges over all subsets of $\{1^2,2^2,\dots,100^2\}$.
[i]Proposed by Connor Gordon and Nairit Sarkar[/i]
Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that:
$$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$
For an integer $n \ge 2$, let $G_n$ be an $n \times n$ grid of unit cells. A subset of cells $H \subseteq G_n$ is considered \textit{quasi-complete} if and only if each row of $G_n$ has at least one cell in $H$ and each column of $G_n$ has at least one cell in $H$. A subset of cells $K \subseteq G_n$ is considered \textit{quasi-perfect} if and only if there is a proper subset $L \subset K$ such that $|L| = n$ and no two elements in $L$ are in the same row or column.
Let $\vartheta(n)$ be the smallest positive integer such that every quasi-complete subset $H \subseteq G_n$ with $|H| \ge \vartheta(n)$ is also quasi-perfect. Moreover, let $\varrho(n)$ be the number of quasi-complete subsets $H \subseteq G_n$ with $|H| = \vartheta(n) - 1$ such that $H$ is not quasi-perfect. Compute $\vartheta(20) + \varrho(20)$.
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.
Find the fundamental group of the topology of $ \text{SL}_2\left(\mathbb{R}\right) $ on $ \mathbb{R}^4. $
Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.
Triangle $GRT$ has $GR=5,$ $RT=12,$ and $GT=13.$ The perpendicular bisector of $GT$ intersects the extension of $GR$ at $O.$ Find $TO.$
[b]a)[/b] Prove that, for any integer $ k, $ the equation $ x^3-24x+k=0 $ has at most an integer solution.
[b]b)[/b] Show that the equation $ x^3+24x-2016=0 $ has exactly one integer solution.
Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.
Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.
Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$
Let $ABC$ be an acute triangle with $AB<AC$, and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$. Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$. Prove that $\angle PBA=\angle PCA$.
[i]Calvin Deng.[/i]
Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$
[b]p1.[/b] Cities A, B, C, D and E are located next to each other along the highway at a distance of $5$ km from each other. The bus runs along the highway from city A to city E and back. The bus consumes $20$ liters of gasoline for every $100$ kilometers. In which city will a bus run out of gas if it initially had $150$ liters of gasoline in its tank?
[b]p2.[/b] Find the minimum four-digit number whose product of all digits is $729$. Explain your answer.
[b]p3.[/b] At the parade, soldiers are lined up in two lines of equal length, and in the first line the distance between adjacent soldiers is $ 20\%$ greater than in the second (there is the same distance between adjacent soldiers in the same line). How many soldiers are in the first rank if there are $85$ soldiers in the second rank?
[b]p4.[/b] It is known about three numbers that the sum of any two of them is not less than twice the third number, and the sum of all three is equal to $300$. Find all triplets of such (not necessarily integer) numbers.
[b]p5.[/b] The tourist fills two tanks of water using two hoses. $2.9$ liters of water flow out per minute from the first hose, $8.7$ liters from the second. At that moment, when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the capacity of the smaller one is $12.5$ liters?
[b]p6.[/b] Is it possible to mark 6 points on a plane and connect them with non-intersecting segments (with ends at these points) so that exactly four segments come out of each point?
[b]p7.[/b] Petya wrote all the natural numbers from $1$ to $1000$ and circled those that are represented as the difference of the squares of two integers. Among the circled numbers, which numbers are more even or odd?
[b]p8.[/b] On a sheet of checkered paper, draw a circle of maximum radius that intersects the grid lines only at the nodes. Explain your answer.
[b]p9.[/b] Along the railway there are kilometer posts at a distance of $1$ km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is $14$ km. What is the maximum distance between the red pillars?
[b]p10.[/b] The island nation is located on $100$ islands connected by bridges, with some islands also connected to the mainland by a bridge. It is known that from each island you can travel to each (possibly through other islands). In order to improve traffic safety, one-way traffic was introduced on all bridges. It turned out that from each island you can leave only one bridge and that from at least one of the islands you can go to the mainland. Prove that from each island you can get to the mainland, and along a single route.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
Calculate the following indefinite integrals.
[1] $\int \ln (x^2-1)dx$
[2] $\int \frac{1}{e^x+1}dx$
[3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$
[4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$
[5] $\int \sqrt{1-\sin x}dx$
Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.)
[i]Proposed by Ankan Bhattacharya, USA[/i]
Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.