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.

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Found problems: 85335

Cono Sur Shortlist - geometry, 2009.G4
Tags: altitude , geometry , perpendicular , incenter
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

1987 Mexico National Olympiad, 1
Tags: fraction , number theory
Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

2019 China Western Mathematical Olympiad, 7
Tags: prime number , number theory
Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

2000 Moldova Team Selection Test, 9
Tags: perfect square , sequence , number theory with sequences , integer
The sequence $x_{n}$ is de fined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

1988 IMO Longlists, 41
Tags: algebra
[b]i.)[/b] Calculate $x$ if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \] [b]ii.)[/b] For each positive number $x,$ let \[ k = \frac{\left( x + \frac{1}{x} \right)^6 - \left( x^6 + \frac{1}{x^6} \right) - 2}{\left( x + \frac{1}{x} \right)^3 - \left( x^3 + \frac{1}{x^3} \right)} \] Calculate the minimum value of $k.$

2014 NIMO Summer Contest, 13
Tags:
Let $\alpha$ and $\beta$ be nonnegative integers. Suppose the number of strictly increasing sequences of integers $a_0,a_1,\dots,a_{2014}$ satisfying $0 \leq a_m \leq 3m$ is $2^\alpha (2\beta + 1)$. Find $\alpha$. [i]Proposed by Lewis Chen[/i]

2003 National Olympiad First Round, 34
Tags:
If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, $\cdots$, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None of the preceding} $

2011 JBMO Shortlist, 5
Tags: combinatorics , set
A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$.

2004 AMC 12/AHSME, 2
Tags:
In the expression $ c\cdot a^b\minus{}d$, the values of $ a$, $ b$, $ c$, and $ d$ are $ 0$, $ 1$, $ 2$, and $ 3$, although not necessarily in that order. What is the maximum possible value of the result? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

2019 AIME Problems, 11
Tags: geometry , excircle
In $\triangle ABC$, the sides have integers lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An [i]excircle[/i] of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$, and the other two excircles are both externally tangent to $\omega$. Find the minimum possible value of the perimeter of $\triangle ABC$.

1999 AMC 8, 6
Tags:
Bo, Coe, Flo, Joe, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe have more than Moe. Jo has more than Moe, but less than Bo. Who has the least amount of money? $ \text{(A)}\ \text{Bo}\qquad\text{(B)}\ \text{Coe}\qquad\text{(C)}\ \text{Flo}\qquad\text{(D)}\ \text{Joe}\qquad\text{(E)}\ \text{Moe} $

2024 Singapore MO Open, Q5
Tags: prime , grid , prime number , combinatorics , number theory
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2012-2013 SDML (Middle School), 7
Tags:
Solve for $x$. $$\frac{1}{2-\frac{3}{4-x}}=5$$ $\text{(A) }-1\qquad\text{(B) }2\qquad\text{(C) }\frac{4}{3}\qquad\text{(D) }\frac{7}{3}\qquad\text{(E) }\frac{13}{5}$

2017 Switzerland - Final Round, 2
Tags: algebra , functional , functional equation
Find all functions f : $R \to R $such that for all $x, y \in R$: $$f(x + yf(x)) = f(xf(y)) - x + f(y + f(x)).$$

2016 May Olympiad, 1
Tags: digit , number theory
We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

2019 LIMIT Category C, Problem 6
Tags: abstract algebra
Which of the following are true? $\textbf{(A)}~GL(n,\mathbb R)\text{ is connected}$ $\textbf{(B)}~GL(n,\mathbb C)\text{ is connected}$ $\textbf{(C)}~O(n,\mathbb R)\text{ is connected}$ $\textbf{(D)}~O(n,\mathbb C)\text{ is connected}$

2024 May Olympiad, 2
Tags: number theory
A number is [i]special[/i] if its tens digit is $9$. For example, $499$ and $1092$ are special, but $509$ is not. Diego has several cards. On each of them, he wrote a special number (he may write the same number on more than one card). When he adds up the numbers on the cards, the total is $2024$. What is the smallest number of cards Diego can have?

2021 Junior Balkan Team Selection Tests - Romania, P3
Tags: incenter , geometry
The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.

2017 Spain Mathematical Olympiad, 4
Tags: combinatorial game , combinatorics
You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?

2011 AIME Problems, 8
Tags: polynomial , reflection , trigonometry , algebra , analytic geometry , geometric transformation , geometry
Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

2015 Chile TST Ibero, 1
Tags: algebra , function
Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

2000 Belarus Team Selection Test, 2.2
Tags:
Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.

2018 CMIMC Algebra, 2
Tags: algebra
Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?

2008 India Regional Mathematical Olympiad, 3
Tags: derivative , vieta , algebra , calculus , inequalities , trigonometry
Suppose $ a$ and $ b$ are real numbers such that the roots of the cubic equation $ ax^3\minus{}x^2\plus{}bx\minus{}1$ are positive real numbers. Prove that: \[ (i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3} \] [19 points out of 100 for the 6 problems]

1993 AMC 12/AHSME, 1
Tags:
For integers $a, b$ and $c$, define $\boxed{a, b, c}$ to mean $a^b-b^c+c^a$. Then $\boxed{1, -1, 2}$ equals $ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $