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2024-03-28T22:07:42Z
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https://artofproblemsolving.com/wiki/index.php?title=L%27H%F4pital%27s_Rule&diff=44124
2012-01-03T03:10:56Z
<p>-Elixir-: /* Discovered By */</p>
<hr />
<div>==Discovered By==<br />
<br />
Named after Guillaume de l'Hopital; however, it is believed that the rule was actually discovered by Johann Bernoulli.<br />
<br />
==The Rule==<br />
<br />
For <math> \frac {0}{0} </math> or <math> \frac {\infty}{\infty} </math> case, the limit <cmath> \lim_{x \to a} \cfrac {f(x)}{g(x)} = \lim_{x \to a} \cfrac {f'(x)}{g'(x)} </cmath> where <math> f'(x) </math> and <math> g'(x) </math> are the first derivatives of <math> f(x) </math> and <math> g(x) </math>, respectively. <br />
<br />
==Examples==<br />
<br />
<math> \lim_{x \to 4} \cfrac {x^3 - 64}{4 - x} = \lim_{x \to 4} \cfrac {3x^2}{-1} = -3(4)^2 = -3(16) = \boxed {-48} </math><br />
<br />
<math> \lim_{x \to 0} \cfrac {\sin x}{x} = \lim_{x \to 0} \cfrac {\cos x}{1} = \cos (0) = \boxed {1} </math><br />
<br />
==Proof of l'Hôpital's rule==<br />
A standard proof of l'Hôpital's rule uses [[Cauchy's mean value theorem]]. l'Hôpital's rule has many variations depending on whether ''c'' and ''L'' are finite or infinite, whether ''f'' and ''g'' converge to zero or infinity, and whether the limits are one-sided or two-sided. <br />
===Zero over zero===<br />
Suppose that ''c'' and ''L'' are finite and ''f'' and ''g'' converge to zero.<br />
<br />
First, define (or redefine) ''f''(''c'')&nbsp;=&nbsp;0 and ''g''(''c'')&nbsp;=&nbsp;0. This makes ''f'' and ''g'' continuous at ''c'', but does not change the limit (since, by definition, the limit does not depend on the value at the point ''c''). <br />
<br />
Since <math>\lim_{x\to c}\frac {f'(x)}{g'(x)}</math> exists, there is an interval (''c''&nbsp;&minus;&nbsp;''δ'',&nbsp;''c''&nbsp;+&nbsp;''δ'') such that for all ''x'' in the interval, with the possible exception of ''x''&nbsp;=&nbsp;''c'', both <math> f'(x)</math> and <math>g'(x)</math> exist and <math>g'(x)</math> is not zero.<br />
<br />
If ''x'' is in the interval (''c'',&nbsp;''c''&nbsp;+&nbsp;''δ''), then the [[mean value theorem]] and Cauchy's mean value theorem both apply to the interval [''c'',&nbsp;''x''] (and a similar statement holds for ''x'' in the interval (''c''&nbsp;&minus;&nbsp;''δ'',&nbsp;''c'')). The mean value theorem implies that ''g''(''x'') is not zero (since otherwise there would be a ''y'' in the interval (''c'',&nbsp;''x'') with <math>g'(y)=0</math>). Cauchy's mean value theorem now implies that there is a point ''ξ''<sub>''x''</sub> in (''c'',&nbsp;''x'') such that<br />
<br />
:<math>\frac{f(x)}{g(x)} = \frac{f'(\xi_x)}{g'(\xi_x)}.</math><br />
<br />
If ''x'' approaches ''c'', then ''ξ''<sub>''x''</sub> approaches ''c'' (by the [[squeeze theorem]] ). Since <math>\lim_{x\to c}f'(x)/g'(x)</math> exists, it follows that<br />
<br />
:<math><br />
\lim_{x\to c}\frac{f(x)}{g(x)}<br />
= \lim_{x\to c}\frac{f'(\xi_x)}{g'(\xi_x)}<br />
= \lim_{x\to c}\frac{f'(x)}{g'(x)}.<br />
</math><br />
<br />
===Infinity over infinity===<br />
Suppose that ''L'' is finite, ''c'' is positive infinity, and ''f'' and ''g'' converge to positive infinity.<br />
<br />
For every ''ε''&nbsp;>&nbsp;0, there is an ''m'' such that<br />
<br />
:<math>\left|\frac{f'(x)}{g'(x)} - L\right| < \varepsilon \quad \text{for } x\geq m.</math><br />
<br />
The mean value theorem implies that if ''x''&nbsp;>&nbsp;''m'', then ''g''(''x'')&nbsp;≠&nbsp;''g''(''m'') (since otherwise there would be a ''y'' in the interval (''m'',&nbsp;''x'') with <math>g'(y)=0</math>). [[Cauchy's mean value theorem]] applied to the interval [''m'',&nbsp;''x''] now implies that<br />
<br />
:<math>\left|\frac{f(x)-f(m)}{g(x)-g(m)} - L\right| < \varepsilon \quad \text{for } x>m.</math><br />
<br />
Since ''f'' converges to positive infinity, if ''x'' is large enough, then ''f''(''x'')&nbsp;≠&nbsp;''f''(''m''). Write<br />
<br />
:<math>\frac{f(x)}{g(x)} = \frac{f(x)-f(m)}{g(x)-g(m)} \cdot \frac{f(x)}{f(x)-f(m)} \cdot \frac{g(x)-g(m)}{g(x)}.</math><br />
<br />
Now,<br />
<br />
:<math><br />
\begin{align}<br />
& \left|\frac{f(x)-f(m)}{g(x)-g(m)} \cdot \frac{f(x)}{f(x)-f(m)} \cdot \frac{g(x)-g(m)}{g(x)} - \frac{f(x)-f(m)}{g(x)-g(m)}\right| \\<br />
& \quad \leq \left|\frac{f(x)-f(m)}{g(x)-g(m)}\right| \left|\frac{f(x)}{f(x)-f(m)} \cdot \frac{g(x)-g(m)}{g(x)} - 1\right| \\<br />
& \quad < (|L|+\varepsilon)\left|\frac{f(x)}{f(x)-f(m)} \cdot \frac{g(x)-g(m)}{g(x)} - 1\right|.<br />
\end{align}<br />
</math><br />
<br />
For ''x'' sufficiently large, this is less than ''ε'' and therefore<br />
<br />
:<math>\left|\frac{f(x)}{g(x)} - L\right| < 2\varepsilon.</math>*<br />
* Note: Steps are missing.</div>
-Elixir-
https://artofproblemsolving.com/wiki/index.php?title=IMO_Problems_and_Solutions&diff=35584
IMO Problems and Solutions
2010-08-08T08:11:19Z
<p>-Elixir-: Fixed spelling. :D</p>
<hr />
<div>This page lists the authors and the proposing countries of the problems of the [[International Mathematical Olympiad|IMO]].<br />
<br />
== 1959 ==<br />
<br />
* [[1959 IMO]] (in Romania)<br />
* [[1959 IMO Problems]]<br />
* [[1959 IMO Problems/Problem 1 | Problem 1]] proposed by Poland<br />
* [[1959 IMO Problems/Problem 2 | Problem 2]] proposed by Constantin Ionescu-Tiu, Romania<br />
* [[1959 IMO Problems/Problem 3 | Problem 3]] proposed by Hungary<br />
* [[1959 IMO Problems/Problem 4 | Problem 4]] proposed by Hungary<br />
* [[1959 IMO Problems/Problem 5 | Problem 5]] proposed by Cezar Cosnita, Romania<br />
* [[1959 IMO Problems/Problem 6 | Problem 6]] proposed by Czechoslovakia<br />
<br />
== 1960 ==<br />
<br />
* [[1960 IMO]] (in Romania)<br />
* [[1960 IMO Problems]]<br />
* [[1960 IMO Problems/Problem 1 | Problem 1]] proposed by Bulgaria<br />
* [[1960 IMO Problems/Problem 2 | Problem 2]] proposed by Hungary<br />
* [[1960 IMO Problems/Problem 3 | Problem 3]] proposed by Gheorghe D. Simionescu, Romania<br />
* [[1960 IMO Problems/Problem 4 | Problem 4]] proposed by Hungary<br />
* [[1960 IMO Problems/Problem 5 | Problem 5]] proposed by Czechoslovakia<br />
* [[1960 IMO Problems/Problem 6 | Problem 6]] proposed by Bulgaria<br />
* [[1960 IMO Problems/Problem 7 | Problem 7]] proposed by East Germany<br />
<br />
== 1961 ==<br />
<br />
* [[1961 IMO]] (in Hungary)<br />
* [[1961 IMO Problems]]<br />
* [[1961 IMO Problems/Problem 1 | Problem 1]] proposed by Hungary<br />
* [[1961 IMO Problems/Problem 2 | Problem 2]] proposed by Poland<br />
* [[1961 IMO Problems/Problem 3 | Problem 3]] proposed by Bulgaria<br />
* [[1961 IMO Problems/Problem 4 | Problem 4]] proposed by East Germany<br />
* [[1961 IMO Problems/Problem 5 | Problem 5]] proposed by Czechoslovakia<br />
* [[1961 IMO Problems/Problem 6 | Problem 6]] proposed by Gheorghe D. Simionescu, Romania<br />
<br />
== 1962 ==<br />
<br />
* [[1962 IMO]] (in Czechoslovakia)<br />
* [[1962 IMO Problems]]<br />
* [[1962 IMO Problems/Problem 1 | Problem 1]] proposed by Poland<br />
* [[1962 IMO Problems/Problem 2 | Problem 2]] proposed by Hungary<br />
* [[1962 IMO Problems/Problem 3 | Problem 3]] proposed by Czechoslovakia<br />
* [[1962 IMO Problems/Problem 4 | Problem 4]] proposed by Cezar Cosnita, Romania<br />
* [[1962 IMO Problems/Problem 5 | Problem 5]] proposed by Bulgaria<br />
* [[1962 IMO Problems/Problem 6 | Problem 6]] proposed by East Germany<br />
* [[1962 IMO Problems/Problem 7 | Problem 7]] proposed by USSR<br />
<br />
== 1963 ==<br />
<br />
* [[1963 IMO]] (in Poland)<br />
* [[1963 IMO Problems]]<br />
* [[1963 IMO Problems/Problem 1 | Problem 1]] proposed by Czechoslovakia<br />
* [[1963 IMO Problems/Problem 2 | Problem 2]] proposed by USSR<br />
* [[1963 IMO Problems/Problem 3 | Problem 3]] proposed by Hungary<br />
* [[1963 IMO Problems/Problem 4 | Problem 4]] proposed by USSR<br />
* [[1963 IMO Problems/Problem 5 | Problem 5]] proposed by Wolfgang Engel, East Germany<br />
* [[1963 IMO Problems/Problem 6 | Problem 6]] proposed by Hungary<br />
<br />
== 1964 ==<br />
<br />
* [[1964 IMO]] (in USSR)<br />
* [[1964 IMO Problems]]<br />
* [[1964 IMO Problems/Problem 1 | Problem 1]] proposed by Czechoslovakia<br />
* [[1964 IMO Problems/Problem 2 | Problem 2]] proposed by Hungary<br />
* [[1964 IMO Problems/Problem 3 | Problem 3]] proposed by Yugoslavia<br />
* [[1964 IMO Problems/Problem 4 | Problem 4]] proposed by Hungary<br />
* [[1964 IMO Problems/Problem 5 | Problem 5]] proposed by Gheorghe D. Simionescu, Romania<br />
* [[1964 IMO Problems/Problem 6 | Problem 6]] proposed by Poland<br />
<br />
== 1965 ==<br />
<br />
* [[1965 IMO]] (in East Germany)<br />
* [[1965 IMO Problems]]<br />
* [[1965 IMO Problems/Problem 1 | Problem 1]] proposed by Yugoslavia<br />
* [[1965 IMO Problems/Problem 2 | Problem 2]] proposed by Poland<br />
* [[1965 IMO Problems/Problem 3 | Problem 3]] proposed by Czechoslovakia<br />
* [[1965 IMO Problems/Problem 4 | Problem 4]] proposed by USSR<br />
* [[1965 IMO Problems/Problem 5 | Problem 5]] proposed by Gheorghe D. Simionescu, Romania<br />
* [[1965 IMO Problems/Problem 6 | Problem 6]] proposed by Poland<br />
<br />
== 1966 ==<br />
<br />
* [[1966 IMO]] (in Bulgaria)<br />
* [[1966 IMO Problems]]<br />
* [[1966 IMO Problems/Problem 1 | Problem 1]] proposed by USSR<br />
* [[1966 IMO Problems/Problem 2 | Problem 2]] proposed by Hungary<br />
* [[1966 IMO Problems/Problem 3 | Problem 3]] proposed by Bulgaria<br />
* [[1966 IMO Problems/Problem 4 | Problem 4]] proposed by Yugoslavia<br />
* [[1966 IMO Problems/Problem 5 | Problem 5]] proposed by Czechoslovakia<br />
* [[1966 IMO Problems/Problem 6 | Problem 6]] proposed by Poland<br />
<br />
== 1967 ==<br />
<br />
* [[1967 IMO]] (in Yugoslavia)<br />
* [[1967 IMO Problems]]<br />
* [[1967 IMO Problems/Problem 1 | Problem 1]] proposed by Poland<br />
* [[1967 IMO Problems/Problem 2 | Problem 2]] proposed by Czechoslovakia<br />
* [[1967 IMO Problems/Problem 3 | Problem 3]] proposed by United Kingdom<br />
* [[1967 IMO Problems/Problem 4 | Problem 4]] proposed by Tullio Viola, Italy<br />
* [[1967 IMO Problems/Problem 5 | Problem 5]] proposed by USSR<br />
* [[1967 IMO Problems/Problem 6 | Problem 6]] proposed by Hungary<br />
<br />
== 1968 ==<br />
<br />
* [[1968 IMO]] (in USSR)<br />
* [[1968 IMO Problems]]<br />
* [[1968 IMO Problems/Problem 1 | Problem 1]] proposed by Unknown Author, Romania (taken from a Romanian problem book)<br />
* [[1968 IMO Problems/Problem 2 | Problem 2]] proposed by Czechoslovakia<br />
* [[1968 IMO Problems/Problem 3 | Problem 3]] proposed by Bulgaria<br />
* [[1968 IMO Problems/Problem 4 | Problem 4]] proposed by Poland<br />
* [[1968 IMO Problems/Problem 5 | Problem 5]] proposed by East Germany<br />
* [[1968 IMO Problems/Problem 6 | Problem 6]] proposed by David Monk, United Kingdom<br />
<br />
== 1969 ==<br />
<br />
* [[1969 IMO]] (in Romania)<br />
* [[1969 IMO Problems]]<br />
* [[1969 IMO Problems/Problem 1 | Problem 1]] proposed by Wolfgang Engel, East Germany<br />
* [[1969 IMO Problems/Problem 2 | Problem 2]] proposed by Hungary<br />
* [[1969 IMO Problems/Problem 3 | Problem 3]] proposed by Poland<br />
* [[1969 IMO Problems/Problem 4 | Problem 4]] proposed by Netherlands<br />
* [[1969 IMO Problems/Problem 5 | Problem 5]] proposed by A. Mekei, Mongolia<br />
* [[1969 IMO Problems/Problem 6 | Problem 6]] proposed by USSR<br />
<br />
== 1970 ==<br />
<br />
* [[1970 IMO]] (in Hungary)<br />
* [[1970 IMO Problems]]<br />
* [[1970 IMO Problems/Problem 1 | Problem 1]] proposed by Poland<br />
* [[1970 IMO Problems/Problem 2 | Problem 2]] proposed by Dumitru Batinetzu, Romania<br />
* [[1970 IMO Problems/Problem 3 | Problem 3]] proposed by Åke Samuelsson, Sweden<br />
* [[1970 IMO Problems/Problem 4 | Problem 4]] proposed by Czechoslovakia<br />
* [[1970 IMO Problems/Problem 5 | Problem 5]] proposed by Bulgaria<br />
* [[1970 IMO Problems/Problem 6 | Problem 6]] proposed by USSR<br />
<br />
== 1971 ==<br />
<br />
* [[1971 IMO]] (in Czechoslovakia)<br />
* [[1971 IMO Problems]]<br />
* [[1971 IMO Problems/Problem 1 | Problem 1]] proposed by Hungary<br />
* [[1971 IMO Problems/Problem 2 | Problem 2]] proposed by USSR<br />
* [[1971 IMO Problems/Problem 3 | Problem 3]] proposed by Poland<br />
* [[1971 IMO Problems/Problem 4 | Problem 4]] proposed by Netherlands<br />
* [[1971 IMO Problems/Problem 5 | Problem 5]] proposed by Bulgaria<br />
* [[1971 IMO Problems/Problem 6 | Problem 6]] proposed by Sweden<br />
<br />
== 1972 ==<br />
<br />
* [[1972 IMO]] (in Poland)<br />
* [[1972 IMO Problems]]<br />
* [[1972 IMO Problems/Problem 1 | Problem 1]] proposed by USSR<br />
* [[1972 IMO Problems/Problem 2 | Problem 2]] proposed by Netherlands<br />
* [[1972 IMO Problems/Problem 3 | Problem 3]] proposed by United Kingdom<br />
* [[1972 IMO Problems/Problem 4 | Problem 4]] proposed by Netherlands<br />
* [[1972 IMO Problems/Problem 5 | Problem 5]] proposed by Bulgaria<br />
* [[1972 IMO Problems/Problem 6 | Problem 6]] proposed by United Kingdom<br />
<br />
== 1973 ==<br />
<br />
* [[1973 IMO]] (in USSR)<br />
* [[1973 IMO Problems]]<br />
* [[1973 IMO Problems/Problem 1 | Problem 1]] proposed by Czechoslovakia<br />
* [[1973 IMO Problems/Problem 2 | Problem 2]] proposed by Poland<br />
* [[1973 IMO Problems/Problem 3 | Problem 3]] proposed by Åke Samuelsson, Sweden<br />
* [[1973 IMO Problems/Problem 4 | Problem 4]] proposed by Yugoslavia<br />
* [[1973 IMO Problems/Problem 5 | Problem 5]] proposed by Poland<br />
* [[1973 IMO Problems/Problem 6 | Problem 6]] proposed by Åke Samuelsson, Sweden<br />
<br />
== 1974 ==<br />
<br />
* [[1974 IMO]] (in East Germany)<br />
* [[1974 IMO Problems]]<br />
* [[1974 IMO Problems/Problem 1 | Problem 1]] proposed by Murray Klamkin, USA<br />
* [[1974 IMO Problems/Problem 2 | Problem 2]] proposed by Matti Lehtinen, Finland<br />
* [[1974 IMO Problems/Problem 3 | Problem 3]] proposed by Ciprian Borcea, Romania<br />
* [[1974 IMO Problems/Problem 4 | Problem 4]] proposed by Bulgaria<br />
* [[1974 IMO Problems/Problem 5 | Problem 5]] proposed by Jan van de Craats, Netherlands<br />
* [[1974 IMO Problems/Problem 6 | Problem 6]] proposed by Sweden<br />
<br />
== 1975 ==<br />
<br />
* [[1975 IMO]] (in Bulgaria)<br />
* [[1975 IMO Problems]]<br />
* [[1975 IMO Problems/Problem 1 | Problem 1]] proposed by Czechoslovakia<br />
* [[1975 IMO Problems/Problem 2 | Problem 2]] proposed by United Kingdom<br />
* [[1975 IMO Problems/Problem 3 | Problem 3]] proposed by Jan van de Craats, Netherlands<br />
* [[1975 IMO Problems/Problem 4 | Problem 4]] proposed by USSR<br />
* [[1975 IMO Problems/Problem 5 | Problem 5]] proposed by USSR<br />
* [[1975 IMO Problems/Problem 6 | Problem 6]] proposed by David Monk, United Kingdom<br />
<br />
== 1976 ==<br />
<br />
* [[1976 IMO]] (in Austria)<br />
* [[1976 IMO Problems]]<br />
* [[1976 IMO Problems/Problem 1 | Problem 1]] proposed by Czechoslovakia<br />
* [[1976 IMO Problems/Problem 2 | Problem 2]] proposed by Matti Lehtinen, Finland<br />
* [[1976 IMO Problems/Problem 3 | Problem 3]] proposed by Jan van de Craats, Netherlands<br />
* [[1976 IMO Problems/Problem 4 | Problem 4]] proposed by Murray Klamkin, USA<br />
* [[1976 IMO Problems/Problem 5 | Problem 5]] proposed by Jan van de Craats, Netherlands<br />
* [[1976 IMO Problems/Problem 6 | Problem 6]] proposed by Robert Lyness and David Monk, United Kingdom<br />
<br />
== 1977 ==<br />
<br />
* [[1977 IMO]] (in Yugoslavia)<br />
* [[1977 IMO Problems]]<br />
* [[1977 IMO Problems/Problem 1 | Problem 1]] proposed by Jan van de Craats, Netherlands<br />
* [[1977 IMO Problems/Problem 2 | Problem 2]] proposed by Phan Duc Chinh, Vietnam<br />
* [[1977 IMO Problems/Problem 3 | Problem 3]] proposed by Jan van de Craats, Netherlands<br />
* [[1977 IMO Problems/Problem 4 | Problem 4]] proposed by David Monk, United Kingdom<br />
* [[1977 IMO Problems/Problem 5 | Problem 5]] proposed by Hermann Frasch, West Germany<br />
* [[1977 IMO Problems/Problem 6 | Problem 6]] proposed by Ivan Prodanov, Bulgaria<br />
<br />
== 1978 ==<br />
<br />
* [[1978 IMO]] (in Romania)<br />
* [[1978 IMO Problems]]<br />
* [[1978 IMO Problems/Problem 1 | Problem 1]] proposed by Cuba<br />
* [[1978 IMO Problems/Problem 2 | Problem 2]] proposed by Murray Klamkin, USA<br />
* [[1978 IMO Problems/Problem 3 | Problem 3]] proposed by United Kingdom<br />
* [[1978 IMO Problems/Problem 4 | Problem 4]] proposed by Murray Klamkin, USA<br />
* [[1978 IMO Problems/Problem 5 | Problem 5]] proposed by France<br />
* [[1978 IMO Problems/Problem 6 | Problem 6]] proposed by Jan van de Craats, Netherlands<br />
<br />
== 1979 ==<br />
<br />
* [[1979 IMO]] (in United Kingdom)<br />
* [[1979 IMO Problems]]<br />
* [[1979 IMO Problems/Problem 1 | Problem 1]] proposed by Arthur Engel, West Germany<br />
* [[1979 IMO Problems/Problem 2 | Problem 2]] proposed by Nickolay Khadzhiivanov and Nedjalko Nenov, Bulgaria<br />
* [[1979 IMO Problems/Problem 3 | Problem 3]] proposed by Nikolai Vasil'ev and Igor F. Sharygin, USSR<br />
* [[1979 IMO Problems/Problem 4 | Problem 4]] proposed by Murray Klamkin, USA<br />
* [[1979 IMO Problems/Problem 5 | Problem 5]] proposed by Joe Gillis, Israel<br />
* [[1979 IMO Problems/Problem 6 | Problem 6]] proposed by Arthur Engel, West Germany<br />
<br />
== 1980 ==<br />
<br />
[[1980 IMO | There was no IMO in 1980]]<br />
<br />
== 1981 ==<br />
<br />
* [[1981 IMO]] (in USA)<br />
* [[1981 IMO Problems]]<br />
* [[1981 IMO Problems/Problem 1 | Problem 1]] proposed by David Monk, United Kingdom<br />
* [[1981 IMO Problems/Problem 2 | Problem 2]] proposed by Arthur Engel, West Germany<br />
* [[1981 IMO Problems/Problem 3 | Problem 3]] proposed by Jan van de Craats, Netherlands<br />
* [[1981 IMO Problems/Problem 4 | Problem 4]] proposed by Belgium<br />
* [[1981 IMO Problems/Problem 5 | Problem 5]] proposed by USSR<br />
* [[1981 IMO Problems/Problem 6 | Problem 6]] proposed by Juha Oikkonen, Finland<br />
<br />
== 1982 ==<br />
<br />
* [[1982 IMO]] (in Hungary)<br />
* [[1982 IMO Problems]]<br />
* [[1982 IMO Problems/Problem 1 | Problem 1]] proposed by David Monk, United Kingdom<br />
* [[1982 IMO Problems/Problem 2 | Problem 2]] proposed by Jan van de Craats, Netherlands<br />
* [[1982 IMO Problems/Problem 3 | Problem 3]] proposed by Alexander N. Grishkov, USSR<br />
* [[1982 IMO Problems/Problem 4 | Problem 4]] proposed by David Monk, United Kingdom<br />
* [[1982 IMO Problems/Problem 5 | Problem 5]] proposed by Jan van de Craats, Netherlands<br />
* [[1982 IMO Problems/Problem 6 | Problem 6]] proposed by Van Nhu Cuong, Vietnam<br />
<br />
== 1983 ==<br />
<br />
* [[1983 IMO]] (in France)<br />
* [[1983 IMO Problems]]<br />
* [[1983 IMO Problems/Problem 1 | Problem 1]] proposed by David Monk, United Kingdom<br />
* [[1983 IMO Problems/Problem 2 | Problem 2]] proposed by Igor F. Sharygin, USSR<br />
* [[1983 IMO Problems/Problem 3 | Problem 3]] proposed by Arthur Engel, West Germany<br />
* [[1983 IMO Problems/Problem 4 | Problem 4]] proposed by Belgium<br />
* [[1983 IMO Problems/Problem 5 | Problem 5]] proposed by Marcin Kuczma, Poland<br />
* [[1983 IMO Problems/Problem 6 | Problem 6]] proposed by Murray Klamkin, USA<br />
<br />
== 1984 ==<br />
<br />
* [[1984 IMO]] (in Czechoslovakia)<br />
* [[1984 IMO Problems]]<br />
* [[1984 IMO Problems/Problem 1 | Problem 1]] proposed by Michael Stoll and Bruno Haible, West Germany<br />
* [[1984 IMO Problems/Problem 2 | Problem 2]] proposed by Aiko Tiggelaar, Netherlands<br />
* [[1984 IMO Problems/Problem 3 | Problem 3]] proposed by Ioan Tomescu, Romania<br />
* [[1984 IMO Problems/Problem 4 | Problem 4]] proposed by Laurentiu Panaitopol, Romania<br />
* [[1984 IMO Problems/Problem 5 | Problem 5]] proposed by Tserendorj Dashdorj, Mongolia<br />
* [[1984 IMO Problems/Problem 6 | Problem 6]] proposed by Poland<br />
<br />
== 1985 ==<br />
<br />
* [[1985 IMO]] (in Finland)<br />
* [[1985 IMO Problems]]<br />
* [[1985 IMO Problems/Problem 1 | Problem 1]] proposed by Frank Budden, United Kingdom<br />
* [[1985 IMO Problems/Problem 2 | Problem 2]] proposed by George Szekeres, Australia<br />
* [[1985 IMO Problems/Problem 3 | Problem 3]] proposed by Jan van de Craats, Netherlands<br />
* [[1985 IMO Problems/Problem 4 | Problem 4]] proposed by R. Gonchigdorj, Mongolia<br />
* [[1985 IMO Problems/Problem 5 | Problem 5]] proposed by Igor F. Sharygin, USSR<br />
* [[1985 IMO Problems/Problem 6 | Problem 6]] proposed by Åke Samuelsson, Sweden<br />
<br />
== 1986 ==<br />
<br />
* [[1986 IMO]] (in Poland)<br />
* [[1986 IMO Problems]]<br />
* [[1986 IMO Problems/Problem 1 | Problem 1]] proposed by Arthur Engel, West Germany<br />
* [[1986 IMO Problems/Problem 2 | Problem 2]] proposed by Gengzhe Chang and Dongxu Qi, China<br />
* [[1986 IMO Problems/Problem 3 | Problem 3]] proposed by Elias Wegert, East Germany<br />
* [[1986 IMO Problems/Problem 4 | Problem 4]] proposed by Sven Sigurðsson, Iceland<br />
* [[1986 IMO Problems/Problem 5 | Problem 5]] proposed by David Monk, United Kingdom<br />
* [[1986 IMO Problems/Problem 6 | Problem 6]] proposed by Konrad Engel, East Germany<br />
<br />
== 1987 ==<br />
<br />
* [[1987 IMO]] (in Cuba)<br />
* [[1987 IMO Problems]]<br />
* [[1987 IMO Problems/Problem 1 | Problem 1]] proposed by Horst Sewerin, West Germany<br />
* [[1987 IMO Problems/Problem 2 | Problem 2]] proposed by I.A. Kushnir, USSR<br />
* [[1987 IMO Problems/Problem 3 | Problem 3]] proposed by Arthur Engel, West Germany<br />
* [[1987 IMO Problems/Problem 4 | Problem 4]] proposed by Nguyen Minh Duc, Vietnam<br />
* [[1987 IMO Problems/Problem 5 | Problem 5]] proposed by Hans-Dietrich Gronau, East Germany<br />
* [[1987 IMO Problems/Problem 6 | Problem 6]] proposed by Vsevolod F. Lev, USSR<br />
<br />
== 1988 ==<br />
<br />
* [[1988 IMO]] (in Australia)<br />
* [[1988 IMO Problems]]<br />
* [[1988 IMO Problems/Problem 1 | Problem 1]] proposed by Lucien Kieffer, Luxembourg<br />
* [[1988 IMO Problems/Problem 2 | Problem 2]] proposed by Czechoslovakia<br />
* [[1988 IMO Problems/Problem 3 | Problem 3]] proposed by David Monk, United Kingdom<br />
* [[1988 IMO Problems/Problem 4 | Problem 4]] proposed by Finbarr Holland, Ireland<br />
* [[1988 IMO Problems/Problem 5 | Problem 5]] proposed by Dimitris Kontogiannis, Greece<br />
* [[1988 IMO Problems/Problem 6 | Problem 6]] proposed by Stephan Beck, West Germany<br />
<br />
== 1989 ==<br />
<br />
* [[1989 IMO]] (in West Germany)<br />
* [[1989 IMO Problems]]<br />
* [[1989 IMO Problems/Problem 1 | Problem 1]] proposed by Jose Marasigan, Philippines<br />
* [[1989 IMO Problems/Problem 2 | Problem 2]] proposed by Esther Szekeres, Australia<br />
* [[1989 IMO Problems/Problem 3 | Problem 3]] proposed by Harm Derksen, Netherlands<br />
* [[1989 IMO Problems/Problem 4 | Problem 4]] proposed by Eggert Briem, Iceland<br />
* [[1989 IMO Problems/Problem 5 | Problem 5]] proposed by Bernt Lindström, Sweden<br />
* [[1989 IMO Problems/Problem 6 | Problem 6]] proposed by Marcin Kuczma, Poland<br />
<br />
== 1990 ==<br />
<br />
* [[1990 IMO]] (in China)<br />
* [[1990 IMO Problems]]<br />
* [[1990 IMO Problems/Problem 1 | Problem 1]] proposed by C.R. Pranesachar, India<br />
* [[1990 IMO Problems/Problem 2 | Problem 2]] proposed by Pavol Černek, Czechoslovakia<br />
* [[1990 IMO Problems/Problem 3 | Problem 3]] proposed by Laurentiu Panaitopol, Romania<br />
* [[1990 IMO Problems/Problem 4 | Problem 4]] proposed by Albert Erkip, Turkey<br />
* [[1990 IMO Problems/Problem 5 | Problem 5]] proposed by Hagen von Eitzen, West Germany<br />
* [[1990 IMO Problems/Problem 6 | Problem 6]] proposed by Harm Derksen, Netherlands<br />
<br />
== 1991 ==<br />
<br />
* [[1991 IMO]] (in Sweden)<br />
* [[1991 IMO Problems]]<br />
* [[1991 IMO Problems/Problem 1 | Problem 1]] proposed by Arkadii Skopenkov, USSR<br />
* [[1991 IMO Problems/Problem 2 | Problem 2]] proposed by Laurentiu Panaitopol, Romania<br />
* [[1991 IMO Problems/Problem 3 | Problem 3]] proposed by Chengzhang Li, China<br />
* [[1991 IMO Problems/Problem 4 | Problem 4]] proposed by Cecil Rousseau, USA<br />
* [[1991 IMO Problems/Problem 5 | Problem 5]] proposed by Johan Yebbou, France<br />
* [[1991 IMO Problems/Problem 6 | Problem 6]] proposed by Harm Derksen, Netherlands<br />
<br />
== 1992 ==<br />
<br />
* [[1992 IMO]] (in Russia)<br />
* [[1992 IMO Problems]]<br />
* [[1992 IMO Problems/Problem 1 | Problem 1]] proposed by Achim Zulauf, New Zealand<br />
* [[1992 IMO Problems/Problem 2 | Problem 2]] proposed by Belur Jana Venkatachala, India<br />
* [[1992 IMO Problems/Problem 3 | Problem 3]] proposed by Chengzhang Li, China<br />
* [[1992 IMO Problems/Problem 4 | Problem 4]] proposed by Johan Yebbou, France<br />
* [[1992 IMO Problems/Problem 5 | Problem 5]] proposed by Stefano Mortola, Italy<br />
* [[1992 IMO Problems/Problem 6 | Problem 6]] proposed by Tony Gardiner, United Kingdom<br />
<br />
== 1993 ==<br />
<br />
* [[1993 IMO]] (in Turkey)<br />
* [[1993 IMO Problems]]<br />
* [[1993 IMO Problems/Problem 1 | Problem 1]] proposed by Tom Laffey, Ireland<br />
* [[1993 IMO Problems/Problem 2 | Problem 2]] proposed by David Monk, United Kingdom<br />
* [[1993 IMO Problems/Problem 3 | Problem 3]] proposed by Kerkko Luosto, Finland<br />
* [[1993 IMO Problems/Problem 4 | Problem 4]] proposed by Donco Dimovski, Macedonia<br />
* [[1993 IMO Problems/Problem 5 | Problem 5]] proposed by Elias Wegert, Germany<br />
* [[1993 IMO Problems/Problem 6 | Problem 6]] proposed by Nicholas de Bruijn, Netherlands<br />
<br />
== 1994 ==<br />
<br />
* [[1994 IMO]] (in Hong Kong)<br />
* [[1994 IMO Problems]]<br />
* [[1994 IMO Problems/Problem 1 | Problem 1]] proposed by Vincent Lafforgue, France<br />
* [[1994 IMO Problems/Problem 2 | Problem 2]] proposed by Hans Lausch, Australia and Garnik Tonoyan, Armenia (independently)<br />
* [[1994 IMO Problems/Problem 3 | Problem 3]] proposed by Gabriel Istrate, Romania<br />
* [[1994 IMO Problems/Problem 4 | Problem 4]] proposed by Hans Lausch, Australia<br />
* [[1994 IMO Problems/Problem 5 | Problem 5]] proposed by David Monk, United Kingdom<br />
* [[1994 IMO Problems/Problem 6 | Problem 6]] proposed by Kerkko Luosto, Finland<br />
<br />
== 1995 ==<br />
<br />
* [[1995 IMO]] (in Canada)<br />
* [[1995 IMO Problems]]<br />
* [[1995 IMO Problems/Problem 1 | Problem 1]] proposed by Borislav Mihailov, Bulgaria<br />
* [[1995 IMO Problems/Problem 2 | Problem 2]] proposed by Nazar Agakhanov, Russia<br />
* [[1995 IMO Problems/Problem 3 | Problem 3]] proposed by Jaromír Šimša, Czech Republic<br />
* [[1995 IMO Problems/Problem 4 | Problem 4]] proposed by Marcin Kuczma, Poland<br />
* [[1995 IMO Problems/Problem 5 | Problem 5]] proposed by Alastair McNaughton, New Zealand<br />
* [[1995 IMO Problems/Problem 6 | Problem 6]] proposed by Marcin Kuczma, Poland<br />
<br />
== 1996 ==<br />
<br />
* [[1996 IMO]] (in India)<br />
* [[1996 IMO Problems]]<br />
* [[1996 IMO Problems/Problem 1 | Problem 1]] proposed by Kerkko Luosto, Finland<br />
* [[1996 IMO Problems/Problem 2 | Problem 2]] proposed by Jeffrey Grossman, Canada<br />
* [[1996 IMO Problems/Problem 3 | Problem 3]] proposed by Mircea Becheanu, Romania<br />
* [[1996 IMO Problems/Problem 4 | Problem 4]] proposed by Alexander S. Golovanov, Russia<br />
* [[1996 IMO Problems/Problem 5 | Problem 5]] proposed by Nairi Sedrakyan, Armenia<br />
* [[1996 IMO Problems/Problem 6 | Problem 6]] proposed by Laurent Rosaz, France<br />
<br />
== 1997 ==<br />
<br />
* [[1997 IMO]] (in Argentina)<br />
* [[1997 IMO Problems]]<br />
* [[1997 IMO Problems/Problem 1 | Problem 1]] proposed by Igor Akylich and Igor Voronovich, Belarus<br />
* [[1997 IMO Problems/Problem 2 | Problem 2]] proposed by David Monk, United Kingdom<br />
* [[1997 IMO Problems/Problem 3 | Problem 3]] proposed by Oleg Bogopolsky, Russia<br />
* [[1997 IMO Problems/Problem 4 | Problem 4]] proposed by Ebadollah S. Mahmoodian and Mohammad Mahdian, Iran<br />
* [[1997 IMO Problems/Problem 5 | Problem 5]] proposed by Petr Kanovsky, Czech Republic<br />
* [[1997 IMO Problems/Problem 6 | Problem 6]] proposed by Giedrius Alkauskas, Lithuania<br />
<br />
== 1998 ==<br />
<br />
* [[1998 IMO]] (in Taiwan)<br />
* [[1998 IMO Problems]]<br />
* [[1998 IMO Problems/Problem 1 | Problem 1]] proposed by Charles Leytem, Luxembourg<br />
* [[1998 IMO Problems/Problem 2 | Problem 2]] proposed by R.B. Bapat, India<br />
* [[1998 IMO Problems/Problem 3 | Problem 3]] proposed by Igor Voronovich, Belarus<br />
* [[1998 IMO Problems/Problem 4 | Problem 4]] proposed by David Monk, United Kingdom<br />
* [[1998 IMO Problems/Problem 5 | Problem 5]] proposed by Vyacheslav Yasinsky, Ukraine<br />
* [[1998 IMO Problems/Problem 6 | Problem 6]] proposed by Oleg Mushkarov and Nikolai Nikolov, Bulgaria<br />
<br />
== 1999 ==<br />
<br />
* [[1999 IMO]] (in Romania)<br />
* [[1999 IMO Problems]]<br />
* [[1999 IMO Problems/Problem 1 | Problem 1]] proposed by Jan Willemson, Estonia<br />
* [[1999 IMO Problems/Problem 2 | Problem 2]] proposed by Marcin Kuczma, Poland<br />
* [[1999 IMO Problems/Problem 3 | Problem 3]] proposed by Eugenii Barabanov and Igor Voronovich, Belarus<br />
* [[1999 IMO Problems/Problem 4 | Problem 4]] proposed by Liang-Ju Chu, Taiwan<br />
* [[1999 IMO Problems/Problem 5 | Problem 5]] proposed by Pavel Kozhevnikov, Russia<br />
* [[1999 IMO Problems/Problem 6 | Problem 6]] proposed by Tetsuya Ando, Japan<br />
<br />
== 2000 ==<br />
<br />
* [[2000 IMO]] (in South-Korea)<br />
* [[2000 IMO Problems]]<br />
* [[2000 IMO Problems/Problem 1 | Problem 1]] proposed by Sergei Berlov, Russia<br />
* [[2000 IMO Problems/Problem 2 | Problem 2]] proposed by Titu Andreescu, USA<br />
* [[2000 IMO Problems/Problem 3 | Problem 3]] proposed by Sergei Shikh and Igor Voronovich, Belarus<br />
* [[2000 IMO Problems/Problem 4 | Problem 4]] proposed by Sándor Dobos, Hungary<br />
* [[2000 IMO Problems/Problem 5 | Problem 5]] proposed by Valerii Senderov, Russia<br />
* [[2000 IMO Problems/Problem 6 | Problem 6]] proposed by Lev and Tatiana Emelianovii, Russia<br />
<br />
== 2001 ==<br />
<br />
* [[2001 IMO]] (in USA)<br />
* [[2001 IMO Problems]]<br />
* [[2001 IMO Problems/Problem 1 | Problem 1]] proposed by Hojoo Lee, South Korea<br />
* [[2001 IMO Problems/Problem 2 | Problem 2]] proposed by Hojoo Lee, South Korea<br />
* [[2001 IMO Problems/Problem 3 | Problem 3]] proposed by Christian Bey, Germany<br />
* [[2001 IMO Problems/Problem 4 | Problem 4]] proposed by Bill Sands, Canada<br />
* [[2001 IMO Problems/Problem 5 | Problem 5]] proposed by Shay Gueron, Israel. (Also: Problem 2501 in Crux Mathematicorum, February 2000)<br />
* [[2001 IMO Problems/Problem 6 | Problem 6]] proposed by Aleksander Ivanov, Bulgaria<br />
<br />
== 2002 ==<br />
<br />
* [[2002 IMO]] (in United Kingdom)<br />
* [[2002 IMO Problems]]<br />
* [[2002 IMO Problems/Problem 1 | Problem 1]] proposed by Federico Ardila, Colombia<br />
* [[2002 IMO Problems/Problem 2 | Problem 2]] proposed by Hojoo Lee, South Korea<br />
* [[2002 IMO Problems/Problem 3 | Problem 3]] proposed by Laurentiu Panaitopol, Romania<br />
* [[2002 IMO Problems/Problem 4 | Problem 4]] proposed by Mihai Manea, Romania<br />
* [[2002 IMO Problems/Problem 5 | Problem 5]] proposed by Belur Jana Venkatachala, India<br />
* [[2002 IMO Problems/Problem 6 | Problem 6]] proposed by Vyacheslav Yasinsky, Ukraine<br />
<br />
== 2003 ==<br />
<br />
* [[2003 IMO]] (in Japan)<br />
* [[2003 IMO Problems]]<br />
* [[2003 IMO Problems/Problem 1 | Problem 1]] proposed by Carlos Gustavo Moreira, Brazil<br />
* [[2003 IMO Problems/Problem 2 | Problem 2]] proposed by Aleksander Ivanov, Bulgaria<br />
* [[2003 IMO Problems/Problem 3 | Problem 3]] proposed by Waldemar Pompe, Poland<br />
* [[2003 IMO Problems/Problem 4 | Problem 4]] proposed by Matti Lehtinen, Finland<br />
* [[2003 IMO Problems/Problem 5 | Problem 5]] proposed by Finbarr Holland, Ireland<br />
* [[2003 IMO Problems/Problem 6 | Problem 6]] proposed by Johan Yebbou, France<br />
<br />
== 2004 ==<br />
<br />
* [[2004 IMO]] (in Greece)<br />
* [[2004 IMO Problems]]<br />
* [[2004 IMO Problems/Problem 1 | Problem 1]] proposed by Dinu Serbanescu and Valentin Vornicu, Romania<br />
* [[2004 IMO Problems/Problem 2 | Problem 2]] proposed by Hojoo Lee, South Korea<br />
* [[2004 IMO Problems/Problem 3 | Problem 3]] proposed by Jan Willemson, Estonia<br />
* [[2004 IMO Problems/Problem 4 | Problem 4]] proposed by Hojoo Lee, South Korea<br />
* [[2004 IMO Problems/Problem 5 | Problem 5]] proposed by Waldemar Pompe, Poland<br />
* [[2004 IMO Problems/Problem 6 | Problem 6]] proposed by Mohsen Jamaali and Armin Morabbi, Iran<br />
<br />
== 2005 ==<br />
<br />
* [[2005 IMO]] (in Mexico)<br />
* [[2005 IMO Problems]]<br />
* [[2005 IMO Problems/Problem 1 | Problem 1]] proposed by Bogdan Enescu, Romania<br />
* [[2005 IMO Problems/Problem 2 | Problem 2]] proposed by Nicholas de Bruijn, Netherlands<br />
* [[2005 IMO Problems/Problem 3 | Problem 3]] proposed by Hojoo Lee, South Korea<br />
* [[2005 IMO Problems/Problem 4 | Problem 4]] proposed by Mariusz Skalba, Poland<br />
* [[2005 IMO Problems/Problem 5 | Problem 5]] proposed by Waldemar Pompe, Poland<br />
* [[2005 IMO Problems/Problem 6 | Problem 6]] proposed by Radu Gologan and Dan Schwarz, Romania<br />
<br />
== 2006 ==<br />
<br />
* [[2006 IMO]] (in Slovenia)<br />
* [[2006 IMO Problems]]<br />
* [[2006 IMO Problems/Problem 1 | Problem 1]] proposed by Hojoo Lee, South Korea<br />
* [[2006 IMO Problems/Problem 2 | Problem 2]] proposed by Dušan Djukić, Serbia<br />
* [[2006 IMO Problems/Problem 3 | Problem 3]] proposed by Finbarr Holland, Ireland<br />
* [[2006 IMO Problems/Problem 4 | Problem 4]] proposed by Zuming Feng, USA<br />
* [[2006 IMO Problems/Problem 5 | Problem 5]] proposed by Dan Schwarz, Romania<br />
* [[2006 IMO Problems/Problem 6 | Problem 6]] proposed by Dušan Djukić, Serbia<br />
<br />
== 2007 ==<br />
<br />
* [[2007 IMO]] (in Vietnam)<br />
* [[2007 IMO Problems]]<br />
* [[2007 IMO Problems/Problem 1 | Problem 1]] proposed by Michael Albert, New Zealand<br />
* [[2007 IMO Problems/Problem 2 | Problem 2]] proposed by Charles Leytem, Luxembourg<br />
* [[2007 IMO Problems/Problem 3 | Problem 3]] proposed by Vasily Astakhov, Russia<br />
* [[2007 IMO Problems/Problem 4 | Problem 4]] proposed by Marek Pechal, Czech Republic<br />
* [[2007 IMO Problems/Problem 5 | Problem 5]] proposed by Kevin Buzzard and Edward Crane, United Kingdom<br />
* [[2007 IMO Problems/Problem 6 | Problem 6]] proposed by Gerhard Woeginger, Netherlands<br />
<br />
== 2008 ==<br />
<br />
* [[2008 IMO]] (in Spain)<br />
* [[2008 IMO Problems]]<br />
* [[2008 IMO Problems/Problem 1 | Problem 1]] proposed by Andrey Gavrilyuk, Russia <br />
* [[2008 IMO Problems/Problem 2 | Problem 2]] proposed by Walther Janous, Austria <br />
* [[2008 IMO Problems/Problem 3 | Problem 3]] proposed by Kęstutis Česnavičius, Lithuania<br />
* [[2008 IMO Problems/Problem 4 | Problem 4]] proposed by Hojoo Lee, South Korea <br />
* [[2008 IMO Problems/Problem 5 | Problem 5]] proposed by Bruno Le Floch and Ilia Smilga, France<br />
* [[2008 IMO Problems/Problem 6 | Problem 6]] proposed by Vladimir Shmarov, Russia<br />
<br />
== 2009 ==<br />
<br />
* [[2009 IMO]] (in Germany)<br />
* [[2009 IMO Problems]]<br />
* [[2009 IMO Problems/Problem 1 | Problem 1]] proposed by Ross Atkins, Australia <br />
* [[2009 IMO Problems/Problem 2 | Problem 2]] proposed by Sergei Berlov, Russia <br />
* [[2009 IMO Problems/Problem 3 | Problem 3]] proposed by Gabriel Carroll, USA<br />
* [[2009 IMO Problems/Problem 4 | Problem 4]] proposed by Hojoo Lee, Peter Vandendriessche and Jan Vonk, Belgium<br />
* [[2009 IMO Problems/Problem 5 | Problem 5]] proposed by Bruno Le Floch, France<br />
* [[2009 IMO Problems/Problem 6 | Problem 6]] proposed by Dmitry Khramtsov, Russia<br />
<br />
== 2010 ==<br />
<br />
* [[2010 IMO]] (in Kazakhstan)<br />
* [[2010 IMO Problems]]<br />
* [[2010 IMO Problems/Problem 1 | Problem 1]] proposed by Pierre Bornsztein, France <br />
* [[2010 IMO Problems/Problem 2 | Problem 2]] proposed by Tai Wai Ming and Wang Chongli, Hong Kong <br />
* [[2010 IMO Problems/Problem 3 | Problem 3]] proposed by Gabriel Carroll, USA <br />
* [[2010 IMO Problems/Problem 4 | Problem 4]] proposed by Poland<br />
* [[2010 IMO Problems/Problem 5 | Problem 5]] proposed by Netherlands<br />
* [[2010 IMO Problems/Problem 6 | Problem 6]] proposed by Morteza Saghafiyan, Iran<br />
<br />
== Comment ==<br />
<br />
This page lists the authors and the proposing countries of the problems of the [[International Mathematical Olympiad|IMO]].<br />
<br />
For many problems, the composers do not have the nationality of the proposing country. <br />
For instance, problem 2009/4 was proposed by Belgium, but only two of the three problem authors were Belgian, whereas the third is from South-Korea.<br />
<br />
To the current moment, there is only a single IMO problem that has two distinct proposing countries: The if-part of problem 1994/2 was proposed by Australia and its only-if part by Armenia. <br />
<br />
== See also ==<br />
<br />
* [[IMO problems statistics]]<br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16 IMO problems on the Resources page]<br />
* [[IMO Shortlist Problems]]<br />
* [[Academic Olympiads]]<br />
* [[Mathematics competition resources]]<br />
<br />
[[Category:Math Contest Problems]]</div>
-Elixir-