Found problems: 85335
2008 Vietnam National Olympiad, 4
he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.
2002 Iran Team Selection Test, 8
We call $A_{1},A_{2},A_{3}$ [i]mangool[/i] iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$. A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$. To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$. Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.
2017 Saudi Arabia BMO TST, 1
Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers
2002 Stanford Mathematics Tournament, 3
An equilateral triangle has has sides $1$ inch long. An ant walks around the triangle maintaining a distance of $1$ inch from the triangle at all times. How far does the ant walk?
2019 MOAA, Speed
[b]p1.[/b] What is $20\times 19 + 20 \div (2 - 7)$?
[b]p2.[/b] Will has three spinners. The first has three equally sized sections numbered $1$, $2$, $3$; the second has four equally sized sections numbered $1$, $2$, $3$, $4$; and the third has five equally sized sections numbered $1$, $2$, $3$, $4$, $5$. When Will spins all three spinners, the probability that the same number appears on all three spinners is $p$. Compute $\frac{1}{p}$.
[b]p3.[/b] Three girls and five boys are seated randomly in a row of eight desks. Let $p$ be the probability that the students at the ends of the row are both boys. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$.
[b]p4.[/b] Jaron either hits a home run or strikes out every time he bats. Last week, his batting average was $.300$. (Jaron's batting average is the number of home runs he has hit divided by the number of times he has batted.) After hitting $10$ home runs and striking out zero times in the last week, Jaron has now raised his batting average to $.310$. How many home runs has Jaron now hit?
[b]p5.[/b] Suppose that the sum $$\frac{1}{1 \cdot 4} +\frac{1}{4 \cdot 7}+ ...+\frac{1}{97 \cdot 100}$$ is expressible as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$.
[b]p6.[/b] Let $ABCD$ be a unit square with center $O$, and $\vartriangle OEF$ be an equilateral triangle with center $A$. Suppose that $M$ is the area of the region inside the square but outside the triangle and $N$ is the area of the region inside the triangle but outside the square, and let $x = |M -N|$ be the positive difference between $M$ and $N$. If $$x =\frac1 8(p -\sqrt{q})$$ for positive integers $p$ and $q$, find $p + q$.
[b]p7.[/b] Find the number of seven-digit numbers such that the sum of any two consecutive digits is divisible by $3$. For example, the number $1212121$ satisfies this property.
[b]p8.[/b] There is a unique positive integer $x$ such that $x^x$ has $703$ positive factors. What is $x$?
[b]p9.[/b] Let $x$ be the number of digits in $2^{2019}$ and let $y$ be the number of digits in $5^{2019}$. Compute $x + y$.
[b]p10.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 13$ and $BC = 10$. Consider the set of all points $D$ in three-dimensional space such that $BCD$ is an equilateral triangle. This set of points forms a circle $\omega$. Let $E$ and $F$ be points on $\omega$ such that $AE$ and $AF$ are tangent to $\omega$. If $EF^2$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, determine $m + n$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Purple Comet Problems, 6
Find $n$ such that $20^{2009}=10^{2000}\cdot 40^9\cdot 2^n$.
2007 Indonesia MO, 7
Points $ A,B,C,D$ are on circle $ S$, such that $ AB$ is the diameter of $ S$, but $ CD$ is not the diameter. Given also that $ C$ and $ D$ are on different sides of $ AB$. The tangents of $ S$ at $ C$ and $ D$ intersect at $ P$. Points $ Q$ and $ R$ are the intersections of line $ AC$ with line $ BD$ and line $ AD$ with line $ BC$, respectively.
(a) Prove that $ P$, $ Q$, and $ R$ are collinear.
(b) Prove that $ QR$ is perpendicular to line $ AB$.
1997 Singapore MO Open, 2
Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.
2021 Sharygin Geometry Olympiad, 5
Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.
2013 Junior Balkan Team Selection Tests - Romania, 4
For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be [i]progressive [/i] if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.
2001 All-Russian Olympiad Regional Round, 10.1
The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.
2012 EGMO, 6
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.)
Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular.
(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.
[i]Romania (Dan Schwarz)[/i]
2020 Durer Math Competition Finals, 4
Endre wrote $n$ (not necessarily distinct) integers on a paper. Then for each of the $2^n$ subsets, Kelemen wrote their sum on the blackboard.
a) For which values of $n$ is it possible that two different $n$-tuples give the same numbers on the blackboard?
b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.
2018 Romanian Master of Mathematics Shortlist, G2
Let $\triangle ABC$ be a triangle, and let $S$ and $T$ be the midpoints of the sides $BC$ and $CA$, respectively. Suppose $M$ is the midpoint of the segment $ST$ and the circle $\omega$ through $A, M$ and $T$ meets the line $AB$ again at $N$. The tangents of $\omega$ at $M$ and $N$ meet at $P$. Prove that $P$ lies on $BC$ if and only if the triangle $ABC$ is isosceles with apex at $A$.
[i]Proposed by Reza Kumara, Indonesia[/i]
2019 Turkey Team SeIection Test, 1
In each one of the given $2019$ boxes, there are $2019$ stones numbered as $1,2,...,2019$ with total mass of $1$ kilogram. In all situations satisfying these conditions, if one can pick stones from different boxes with different numbers, with total mass of at least 1 kilogram, in $k$ different ways, what is the maximal of $k$?
2005 MOP Homework, 1
Given real numbers $x$, $y$, $z$ such that $xyz=-1$, show that
$x^4+y^4+z^4+3(x+y+z) \ge \sum_{sym} \frac{x^2}{y}$.
2022 China Team Selection Test, 5
Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$.
(2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$.
Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.
MathLinks Contest 6th, 5.3
Let $ABC$ be a triangle, and let $ABB_2A_3$, $BCC_3B_1$ and $CAA_1C_2$ be squares constructed outside the triangle. Denote with $S$ the area of the triangle $ABC$ and with s the area of the triangle formed by the intersection of the lines $A_1B_1$, $B_2C_2$ and $C_3A_3$. Prove that $s \le (4 - 2\sqrt3)S$.
2016 Mediterranean Mathematics Olympiad, 2
Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that
\[ \sqrt{\frac{b}{a^2+3}}+
\sqrt{\frac{c}{b^2+3}}+
\sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]
2017 Polish MO Finals, 4
Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.
1946 Putnam, A6
A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle, but the form of the function is not known. A motion is observed, and the distance $x$ covered in time $t$ satisfies the formula $x= at^2 + bt+c$, where $a,b,c$ have numerical values determined by observation of the motion. Find the function $f(v)$ for the range of $v$ covered by the experiment.
2007 District Olympiad, 4
Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$
Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:
2018 Tuymaada Olympiad, 5
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
[i]Proposed by N. Vlasova, S. Berlov[/i]
1991 Spain Mathematical Olympiad, 5
For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.
2023 China MO, 5
Prove that there exist $C>0$, which satisfies the following conclusion:
For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree.
Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$.
[i]Proposed by Qu Zhenhua[/i]