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: 6530

2008 China Girls Math Olympiad, 2

Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that \[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0. \]

2006 Germany Team Selection Test, 2

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

1948 Moscow Mathematical Olympiad, 145

Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$

1964 AMC 12/AHSME, 27

Tags: inequalities
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: $ \textbf{(A)}\ 0<a<.01\qquad\textbf{(B)}\ .01<a<1 \qquad\textbf{(C)}\ 0<a<1\qquad$ $\textbf{(D)}\ 0<a \le 1\qquad\textbf{(E)}\ a>1 $

2014 AMC 10, 7

Tags: inequalities
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true? (I) $x+y < a+b$ (II) $x-y < a-b$ (III) $xy < ab$ (IV) $\frac{x}{y} < \frac{a}{b}$ ${ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$

2010 IMO Shortlist, 8

Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that \[2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.\] [i]Proposed by Sung Yun Kim, South Korea[/i]

2009 USAMTS Problems, 1

Archimedes planned to count all of the prime numbers between $2$ and $1000$ using the Sieve of Eratosthenes as follows: (a) List the integers from $2$ to $1000$. (b) Circle the smallest number in the list and call this $p$. (c) Cross out all multiples of $p$ in the list except for $p$ itself. (d) Let $p$ be the smallest number remaining that is neither circled nor crossed out. Circle $p$. (e) Repeat steps $(c)$ and $(d)$ until each number is either circled or crossed out. At the end of this process, the circled numbers are prime and the crossed out numbers are composite. Unfortunately, while crossing out the multiples of $2$, Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides $2$). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from $2$ to $1000$ (including $2$), then what is the largest possible prime that Archimedes accidentally crossed out?

1996 All-Russian Olympiad, 7

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

1965 AMC 12/AHSME, 13

Let $ n$ be the number of number-pairs $ (x,y)$ which satisfy $ 5y \minus{} 3x \equal{} 15$ and $ x^2 \plus{} y^2 \le 16$. Then $ n$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{more than two, but finite} \qquad \textbf{(E)}\ \text{greater than any finite number}$

2005 Germany Team Selection Test, 2

Tags: inequalities
Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$. Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.

1998 VJIMC, Problem 4-M

Prove the inequality $$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.

1950 Moscow Mathematical Olympiad, 176

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$

2009 China Western Mathematical Olympiad, 4

Tags: inequalities
The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.

2024 5th Memorial "Aleksandar Blazhevski-Cane", P2

Let $x,y$ and $z$ be positive real numbers such that $xy+z^2=8$. Determine the smallest possible value of the expression $$\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.$$

2014 ELMO Shortlist, 6

Tags: inequalities
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]

2009 Sharygin Geometry Olympiad, 9

Given $ n$ points on the plane, which are the vertices of a convex polygon, $ n > 3$. There exists $ k$ regular triangles with the side equal to $ 1$ and the vertices at the given points. [list][*] Prove that $ k < \frac {2}{3}n$. [*] Construct the configuration with $ k > 0.666n$.[/list]

2011 Morocco National Olympiad, 1

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

2023 China Team Selection Test, P21

Tags: inequalities
Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$, $$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$

2022 Junior Balkan Team Selection Tests - Romania, P1

Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$ [list=a] [*]If $a+b+c+d=6,$ prove that $d<0,36.$ [*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold? [/list]

the 6th XMO, 2

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

2003 Kazakhstan National Olympiad, 2

For positive real numbers $ x, y, z $, prove the inequality: $$ \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.} $$

1969 AMC 12/AHSME, 25

If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is: $\textbf{(A) }2\sqrt6\qquad \textbf{(B) }6\qquad \textbf{(C) }8\sqrt2\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{none of these.}$

2016 Grand Duchy of Lithuania, 1

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$$

2006 China Second Round Olympiad, 3

Tags: inequalities
Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $

2012 Grigore Moisil Intercounty, 1

The absolute value of the sum of the elements of a real orthogonal matrix is at most the order of the matrix.