This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 85335

2019 India IMO Training Camp, P3
Tags: algebra
Let $n\ge 2$ be an integer. Solve in reals: \[|a_1-a_2|=2|a_2-a_3|=3|a_3-a_4|=\cdots=n|a_n-a_1|.\]

2021-2022 OMMC, 10
Tags:
A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find \[\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3.\] [i]Proposed by Evan Chang[/i]

2003 India IMO Training Camp, 10
Tags: modular arithmetic , number theory
Let $n$ be a positive integer greater than $1$, and let $p$ be a prime such that $n$ divides $p-1$ and $p$ divides $n^3-1$. Prove that $4p-3$ is a square.

1999 Harvard-MIT Mathematics Tournament, 9
Tags: algebra
Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$

2023 Taiwan TST Round 1, 1
Tags: algebra
Let $\mathbb{Q}_{>1}$ be the set of rational numbers greater than $1$. Let $f:\mathbb{Q}_{>1}\to \mathbb{Z}$ be a function that satisfies \[f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}\] Show that for any $a,b\in\mathbb{Q}_{>1}$ with $\frac{1}{a}+\frac{1}{b}=1$, we have $f(a)+f(b)=-2$. [i]Proposed by usjl[/i]

2017 Hanoi Open Mathematics Competitions, 13
Tags: inequalities , distance , geometric inequalities , geometry
Let $ABC$ be a triangle. For some $d>0$ let $P$ stand for a point inside the triangle such that $|AB| - |P B| \ge d$, and $|AC | - |P C | \ge d$. Is the following inequality true $|AM | - |P M | \ge d$, for any position of $M \in BC $?

2020 Bangladesh Mathematical Olympiad National, Problem 7
Tags: coordinate geometry , algebra , geometry
$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?

2000 IMO Shortlist, 2
Tags: number theory , divisor
For a positive integer $n$, let $d(n)$ be the number of all positive divisors of $n$. Find all positive integers $n$ such that $d(n)^3=4n$.

IMSC 2023, 5
Tags: combinatorics
In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]

2022 Portugal MO, 1
Tags: combinatorics
Raul's class has $15$ students, all with different heights. The Mathematics teacher wants to place them in a queue so that, at the beginning of the queue, they are ordered in ascending order of heights, from then on, they are ordered in descending order and Raul, who He is the tallest in the class, he cannot be at the extremes. In how many different ways is it possible to form this queue?

1988 IberoAmerican, 2
Tags: number theory
Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$. Prove that: $(a)$ $q\ge b+d$ $(b)$ If $q=b+d$, then $p=a+c$.

2013 Bogdan Stan, 4
Tags: seqeunce , cesaro-stolz , real analysis , sequence
Let be a sequence $ \left( x_n \right)_{n\ge 1} $ having the property that $$ \lim_{n\to\infty } \left( 14(n+2)x_{n+2} -15(n+1)x_{n+1} +nx_n \right) =13. $$ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent and calculate its limit. [i]Cosmin Nițu[/i]

2013 BMT Spring, 4
Tags: algebra
Find the sum of all real numbers $x$ such that $x^2 = 5x + 6\sqrt{x} - 3$.

1991 Arnold's Trivium, 57
Tags:
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 $z\to\infty$.

2010 Iran MO (2nd Round), 4
Tags: polynomial , inequalities , algebra
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\] Prove that $P(x)$ do not have a real root in $[-1,1]$.

2013 AMC 10, 8
Tags: algebra
What is the value of \[\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}?\] $ \textbf{(A)}\ -1\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ 2013\qquad\textbf{(E)}\ 2^{4024} $

2013 BMT Spring, 6
Tags: combinatorics , probability
A coin is flipped until there is a head followed by two tails. What is the probability that this will take exactly $12$ flips?

1951 Putnam, B6
Tags:
Assuming that all of the roots of the cubic equation $x^3 + ax^2 +bx + c = 0$ are real, show that the difference between the greatest and the least roots is not less than $(a^2 - 3b)^{1/2}$ or greater than $2 (a^2 - 3b)^{1/2} / 3^{1/2}.$

2013 Abels Math Contest (Norwegian MO) Final, 4a
Tags: combinatorics
An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called [i]crossing [/i] if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many [i]crossing [/i] quadruples are there in the $2013$-gon?

2010 Math Prize for Girls Olympiad, 4
Tags: analytic geometry , algorithm , induction
Let $S$ be a set of $n$ points in the coordinate plane. Say that a pair of points is [i]aligned[/i] if the two points have the same $x$-coordinate or $y$-coordinate. Prove that $S$ can be partitioned into disjoint subsets such that (a) each of these subsets is a collinear set of points, and (b) at most $n^{3/2}$ unordered pairs of distinct points in $S$ are aligned but not in the same subset.

1995 Baltic Way, 4
Tags: number theory
Josh is older than Fred. Josh notices that if he switches the two digits of his age (an integer), he gets Fred’s age. Moreover, the difference between the squares of their ages is a square of an integer. How old are Josh and Fred?

2014 Mexico National Olympiad, 5
Tags: holder , inequalities
Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove: \[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \] And determine when equality holds.

BIMO 2022, 6
Tags: geometry
Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

2013 AMC 10, 6
Tags:
Joey and his five brothers are ages $3,5,7,9,11,$ and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 13 $

2025 Portugal MO, 5
Tags: number theory
An integer number $n \geq 2$ is called [i]feirense[/i] if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.