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: 24

2020 Kosovo National Mathematical Olympiad, 1
Tags: power , compare , pow , algebra , number theory
Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.

1973 Chisinau City MO, 69
Tags: compare , algebra
Greater or less than one is the number $0.99999^{1.00001} \cdot 1.00001^{0.99999}$?

1949-56 Chisinau City MO, 38
Tags: compare , algebra
Which is more $\log_3 7$ or $\log_{\frac{1}{3}} \frac{1}{7}$ ?

1982 All Soviet Union Mathematical Olympiad, 335
Tags: algebra , compare , angle , trigonometry
Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.

III Soros Olympiad 1996 - 97 (Russia), 11.3
Tags: compare , algebra , inequalities
Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or $\sqrt[5]{13}$? (Use of a calculator is prohibited.)

1972 Swedish Mathematical Competition, 4
Tags: logarithm , algebra , compare , inequalities
Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$. Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[ 0.47 < \log_{10} 3 < 0.482. \]

2022 Assara - South Russian Girl's MO, 7
Tags: compare , algebra
Find out which of the two numbers is greater: $$\dfrac{2}{2 +\dfrac{2}{2 +\dfrac{2}{... +\dfrac{2}{2+\frac22}}}} \,\,\, \text{or} \,\,\, \dfrac{3}{3 +\dfrac{3}{3 +\dfrac{3}{... +\dfrac{3}{3+\frac33}}}}$$ (Each expression has $2022$ fraction signs.)

2010 Hanoi Open Mathematics Competitions, 1
Tags: algebra , compare
Compare the numbers: $P = 888...888 \times 333 .. 333$ ($2010$ digits of $8$ and $2010$ digits of $3$) and $Q = 444...444\times 666...6667$ ($2010$ digits of $4$ and $2009$ digits of $6$) (A): $P = Q$, (B): $P > Q$, (C): $P < Q$.

1995 Chile National Olympiad, 6
Tags: algebra , compare
Which of the following rationals is greater , $\frac{1995^{1994} + 1}{1995^{1995} + 1}$ or $\frac{1995^{1995} + 1}{ 1995^{1996} +1}$ ?

1984 All Soviet Union Mathematical Olympiad, 392
Tags: algebra , compare , logarithm
What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).

2000 Estonia National Olympiad, 2
Tags: algebra , compare
Which of the numbers $2^{2002}$ and $2000^{200}$ is bigger?

2019 Hanoi Open Mathematics Competitions, 1
Tags: compare , algebra
Let $x$ and $y$ be positive real numbers. Which of the following expressions is larger than both $x$ and $y$? [b]A.[/b] $xy + 1$ [b]B.[/b] $(x + y)^2$ [b]C.[/b] $x^2 + y$ [b]D.[/b] $x(x + y)$ [b]E.[/b] $(x + y + 1)^2$

2016 Argentina National Olympiad, 2
Tags: algebra , sum , compare
For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$ .

2018 Junior Regional Olympiad - FBH, 3
Tags: compare , root
Let $a$, $b$ and $m$ be three positive real numbers and $a>b$. Which of the numbers $A=\sqrt{a+m}-\sqrt{a}$ and $B=\sqrt{b+m}-\sqrt{b}$ is bigger:

1996 Estonia National Olympiad, 2
Tags: algebra , compare
Which number is greater, $\frac{1996^{1995}+1}{1996^{1996}+1}$ or $ \frac{1996^{1996}+1}{1996^{1997}+1}$ ?

1940 Moscow Mathematical Olympiad, 067
Tags: compare , factorial , algebra
Which is greater: $300!$ or $100^{300}$?

2014 Junior Regional Olympiad - FBH, 1
Tags: compare
Compare numbers $A=5+2\sqrt{5}$ and $B=\sqrt{45+20\sqrt{5}}$

1992 Tournament Of Towns, (326) 3
Tags: algebra , inequalities , compare , radical
Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater: $$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$ Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression. (N. Kurlandchik)

2023 Kyiv City MO Round 1, Problem 1
Tags: algebra , compare
Which number is larger: $A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}$, or $B = \log_{2023} 91125$?

2015 BAMO, 3
Tags: fraction , compare , algebra
Which number is larger, $A$ or $B$, where $$A = \dfrac{1}{2015} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2015})$$ and $$B = \dfrac{1}{2016} (1 + \dfrac12 + \dfrac13 + \cdots + \dfrac{1}{2016}) \text{ ?}$$ Prove your answer is correct.

2017 Junior Regional Olympiad - FBH, 3
Tags: angle , triangle , compare
In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$

2011 Sharygin Geometry Olympiad, 17
Tags: geometry , geometric inequality , median , bisector , compare
a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

1951 Moscow Mathematical Olympiad, 190
Tags: algebra , compare
Which number is greater: $\frac{2.00 000 000 004}{(1.00 000 000 004)^2 + 2.00 000 000 004}$ or $\frac{2.00 000 000 002}{(1.00 000 000 002)^2 + 2.00 000 000 002}$ ?

1978 Chisinau City MO, 154
Tags: algebra , radical , compare
What's more $\sqrt[4]{7}+\sqrt[4]{11}$ or $2\sqrt{\frac{\sqrt{7}+\sqrt{11}}{2}}$ ?