Found problems: 85335
2010 Contests, 2
The [i]rank[/i] of a rational number $q$ is the unique $k$ for which $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$, where each $a_i$ is the smallest positive integer $q$ such that $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$. Let $q$ be the largest rational number less than $\frac{1}{4}$ with rank $3$, and suppose the expression for $q$ is $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$. Find the ordered triple $(a_1,a_2,a_3)$.
2008 IberoAmerican, 6
[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.
2024 Baltic Way, 17
Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?
2014 District Olympiad, 2
Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$
Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$
2019 Tournament Of Towns, 2
Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.
2023 Tuymaada Olympiad, 3
Prove that for every positive integer $n \geq 2$, $$\frac{\sum_{1\leq i \leq n} \sqrt[3]{\frac{i}{n+1}}}{n} \leq \frac{\sum_{1\leq i \leq n-1} \sqrt[3]{\frac{i}{n}}}{n-1}.$$
2017 India IMO Training Camp, 2
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2016 India Regional Mathematical Olympiad, 6
$ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1,P_2, \dots, P_10$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \dots, Q_10$ for the side $CA$ and $R_1,R_2,\dots, R_10$ for the side $AB$. Find the number of triples $(i,j,k)$ with $i,j,k \in \{1,2,\dots,10\}$ such that the centroids of triangles $ABC$ and $P_iQ_jR_k$ coincide.
1953 AMC 12/AHSME, 23
The equation $ \sqrt {x \plus{} 10} \minus{} \frac {6}{\sqrt {x \plus{} 10}} \equal{} 5$ has:
$ \textbf{(A)}\ \text{an extraneous root between } \minus{} 5\text{ and } \minus{} 1 \\
\textbf{(B)}\ \text{an extraneous root between } \minus{} 10\text{ and } \minus{} 6 \\
\textbf{(C)}\ \text{a true root between }20\text{ and }25 \qquad\textbf{(D)}\ \text{two true roots} \\
\textbf{(E)}\ \text{two extraneous roots}$
2002 Singapore Senior Math Olympiad, 1
Let $f: N \to N$ be a function satisfying the following:
$\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$.
$\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes.
Determine all possible values of $f(2002)$. Justify your answers.
2006 Sharygin Geometry Olympiad, 8.2
What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?
2008 India National Olympiad, 6
Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that
[b](i)[/b] $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and
[b](ii)[/b] $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.
2020 BMT Fall, 18
Let $x$ and $y$ be integers between $0$ and $5$, inclusive. For the system of modular congruences $$ \begin{cases}
x + 3y \equiv 1 \,\,(mod \, 2) \\
4x + 5y \equiv 2 \,\,(mod \, 3) \end{cases}$$, find the sum of all distinct possible values of $x + y$
1993 China Team Selection Test, 3
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
2013 Middle European Mathematical Olympiad, 4
Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property?
Gheorghe Țițeica 2024, P1
Let $E(x,y)=\frac{(1+x)(1+y)(1+xy)}{(1+x^2)(1+y^2)}$. Find the minimum and maximum value of $E$ on $\mathbb{R}^2$.
[i]Dorel Miheț[/i]
1991 Arnold's Trivium, 57
Find the dimension of the solution space of the problem $\partial u/\partial \overline{z} = \delta(z - i)$ for $\text{Im } z \ge 0$,
$\text{Im } u(z) = 0$ for $\text{Im } z = 0$, $u\to 0$ as $z\to\infty$.
2017 Indonesia Juniors, day 1
p1. Find all real numbers $x$ that satisfy the inequality $$\frac{x^2-3}{x^2-1}+ \frac{x^2 + 5}{x^2 + 3} \ge \frac{x^2-5}{x^2-3}+\frac{x^2 + 3}{x^2 + 1}$$
p2. It is known that $m$ is a four-digit natural number with the same units and thousands digits. If $m$ is a square of an integer, find all possible numbers $m$.
p3. In the following figure, $\vartriangle ABP$ is an isosceles triangle, with $AB = BP$ and point $C$ on $BP$. Calculate the volume of the object obtained by rotating $ \vartriangle ABC$ around the line $AP$
[img]https://cdn.artofproblemsolving.com/attachments/c/a/65157e2d49d0d4f0f087f3732c75d96a49036d.png[/img]
p4. A class farewell event is attended by $10$ male students and $ 12$ female students. Homeroom teacher from the class provides six prizes to randomly selected students. Gifts that provided are one school bag, two novels, and three calculators. If the total students The number of male students who received prizes was equal to the total number of female students who received prizes. How many possible arrangements are there of the student who gets the prize?
p5. It is known that $S =\{1945, 1946, 1947, ..., 2016, 2017\}$. If $A = \{a, b, c, d, e\}$ a subset of $S$ where $a + b + c + d + e$ is divisible by $5$, find the number of possible $A$'s.
Russian TST 2015, P1
The points $A', B', C', D'$ are selected respectively on the sides $AB, BC, CD, DA$ of the cyclic quadrilateral $ABCD$. It is known that $AA' = BB' = CC' = DD'$ and \[\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.\]Prove that $ABCD$ is a square.
2012 Miklós Schweitzer, 6
Let $A,B,C$ be matrices with complex elements such that $[A,B]=C, [B,C]=A$ and $[C,A]=B$, where $[X,Y]$ denotes the $XY-YX$ commutator of the matrices. Prove that $e^{4 \pi A}$ is the identity matrix.
2017 Romania Team Selection Test, P3
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
2018 Harvard-MIT Mathematics Tournament, 5
A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew?
2014 Contests, 1
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2001 Baltic Way, 6
The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.
2012 Pre-Preparation Course Examination, 2
Suppose that $\lim_{n\to \infty} a_n=a$ and $\lim_{n\to \infty} b_n=b$. Prove that
$\lim_{n\to \infty}\frac{1}{n}(a_1b_n+a_2b_{n-1}+...+a_nb_1)=ab$.