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: 6530

1978 Kurschak Competition, 3

A triangle has inradius $r$ and circumradius $R$. Its longest altitude has length $H$. Show that if the triangle does not have an obtuse angle, then $H \ge r+R$. When does equality hold?

2015 China Western Mathematical Olympiad, 3

Tags: inequalities
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be positive real numbers such that $\sum_{i=1}^nx_i=1$ .Prove that$$\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.$$

2018 India IMO Training Camp, 2

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

1963 German National Olympiad, 3

It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds $$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$ Under what conditions does equality occur?

2019 Centroamerican and Caribbean Math Olympiad, 5

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

1990 Irish Math Olympiad, 1

Tags: inequalities
Let $n>3$ be a natural number . Prove that \[\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.\]

2008 Balkan MO, 2

Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$: a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$ b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?

Oliforum Contest I 2008, 3

Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.

2005 Germany Team Selection Test, 3

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

1990 Tournament Of Towns, (251) 5

Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$ (recalling that $ \sqrt2 = 1.414213..$.). (D. Fomin, Leningrad)

2021 JHMT HS, 6

Let $f$ be a function whose domain is $[1, 20]$ and whose range is a subset of $[-100, 100].$ Suppose $\tfrac{f(x)}{y} - \tfrac{f(y)}{x} \leq (x - y)^2$ for all $x$ and $y$ in $[1, 20].$ Compute the largest value of $f(x) - f(y)$ over all such functions $f$ and all $x$ and $y$ in the domain $[1, 20].$

1997 Polish MO Finals, 2

Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}

2022 Kazakhstan National Olympiad, 5

For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that $$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$

1996 AMC 12/AHSME, 25

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have? $\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$

BIMO 2022, 1

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

2013 South africa National Olympiad, 6

Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.

1991 AIME Problems, 6

Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

2022 Irish Math Olympiad, 6

6. Suppose [i]a[/i], [i]b[/i], [i]c[/i] are real numbers such that [i]a[/i] + [i]b[/i] + [i]c[/i] = 1. Prove that \[a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.\]

MathLinks Contest 1st, 1

Tags: inequalities
Prove that for every positive numbers $x, y, z$ the following inequality holds: $$\sqrt{4x^2 + 4x(y + z) + (y - z)^2} <\sqrt{4y^2 + 4y(z + x) + (z - x)^2}+\sqrt{4z^2 + 4z(x + y) + (x - y)^2}.$$

2011 Balkan MO Shortlist, A4

Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that \begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}

2021 Korea National Olympiad, P5

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

1992 Denmark MO - Mohr Contest, 3

Let $x$ and $y$ be positive numbers with $x +y=1$. Show that $$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \ge 9.$$

2018 District Olympiad, 4

a) Consider the positive integers $a, b, c$ so that $a < b < c$ and $a^2+b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2+a_2^2+a_3^2=a_4^2$ and $a_1 < a_2 < a_3 < a_4$. b) Show that for any $n \in N$, $n\ge 3$, there exist the positive integers $a_1, a_2,..., a_n$ so that $a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2$ and $a_1 < a_2 < ...< a_{n-1} < a_n$

1991 Putnam, A5

A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have $\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$ Now what?

1988 IMO Longlists, 64

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$