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Difference between revisions of "2001 IMO Problems"

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Consider an acute triangle <math>\triangle ABC</math>. Let <math>P</math> be the foot of the altitude of triangle <math>\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assume that <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>.
 
Consider an acute triangle <math>\triangle ABC</math>. Let <math>P</math> be the foot of the altitude of triangle <math>\triangle ABC</math> issuing from the vertex <math>A</math>, and let <math>O</math> be the [[circumcenter]] of triangle <math>\triangle ABC</math>. Assume that <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>.
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[[2001 IMO Problems/Problem 1| Solution]]
 
[[2001 IMO Problems/Problem 1| Solution]]
  

Revision as of 18:49, 12 December 2024

Problem 1

Consider an acute triangle $\triangle ABC$. Let $P$ be the foot of the altitude of triangle $\triangle ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $\triangle ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

Solution

Problem 2

Let $a,b,c$ be positive real numbers. Prove that $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$.

Problem 3

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

Problem 4

Let $n_1, n_2, \dots , n_m$ be integers where $m>1$ is odd. Let $x = (x_1, \dots , x_m)$ denote a permutation of the integers $1, 2, \cdots , m$. Let $f(x) = x_1n_1 + x_2n_2 + ... + x_mn_m$. Show that for some distinct permutations $a$, $b$ the difference $f(a) - f(b)$ is a multiple of $m!$.

Problem 5

$ABC$ is a triangle. $X$ lies on $BC$ and $AX$ bisects angle $A$. $Y$ lies on $CA$ and $BY$ bisects angle $B$. Angle $A$ is $60^{\circ}$. $AB + BX = AY + YB$. Find all possible values for angle $B$.

Problem 6

$k > l > m > n$ are positive integers such that $km + ln = (k+l-m+n)(-k+l+m+n)$. Prove that $kl+mn$ is not prime.

See Also

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