[/quote]
,
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
.
,
is a perfect square.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
and 
Prove that
where 
be a function defined on 
taking values in 
such that
for all 
for all 
;
identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables 
a) of degree 
;
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
.
)
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
Y by Adventure10, Mango247
1. Let the 
are positive numbers. Prove that ![\[\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+24\cdot\dfrac{\sqrt[3]{abc}}{a+b+c}\geq 11.\]](//latex.artofproblemsolving.com/1/3/3/133528c378ea6bb72d534414b1dfe3a84e46f777.png)
2. Let
, 
and 
are positives. Prove that ![\[\frac{(a+b)(b+c)(c+a)}{abc}\ge\frac{17(a+b+c)}{3\sqrt[3]{abc}}-9. \]](//latex.artofproblemsolving.com/a/9/e/a9e476212a87aa50470f113d05c946ce3276e616.png)
3. Let, and are positive numbers. Prove that
![\[ \sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}+\sqrt{\frac{a+b}{c}}\ge \sqrt{ 11\cdot\frac{a+b+c}{\sqrt[3]{abc}}-15 }.\]](//latex.artofproblemsolving.com/e/4/3/e43ce425c843181d8f76e8975496a14c6d231213.png)
4. Let, and are positive numbers. Prove that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge\sqrt{15\cdot\frac{a^2+b^2+c^2}{ab+bc+ca}-6}.\]](//latex.artofproblemsolving.com/4/0/d/40db9f980fd659e134477be5690644f4810dbfcc.png)
5. For positives, and prove that
![\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{9abc}{a^2+b^2+c^2}\ge 2(a+b+c).\]](//latex.artofproblemsolving.com/6/c/e/6ceb38df797ff34e4264361ef3fc5d837b7f45a4.png)
6. For positives, and prove that ![\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq\frac{37(a^2+b^2+c^2)-19(ab+ac+bc)}{6(a+b+c)}.\]](//latex.artofproblemsolving.com/a/b/1/ab17eb90ce1d96142dd1daa801135121f48af422.png)
7. For positives, and prove that ![\[(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq9\left(\frac{a^2+b^2+c^2}{ab+ac+bc}\right)^{\frac{4}{7}}.\]](//latex.artofproblemsolving.com/6/c/0/6c092d438e41b1b7c76fddc8fc0ff358dc0086ca.png)
8. Let, and are non-negative numbers such that 
. Prove that
![\[\sqrt a + \sqrt b + \sqrt c \ge \sqrt[7]{{9{{(ab + bc + ca)}^5}}}.\]](//latex.artofproblemsolving.com/1/6/0/16067b3123264797bbb34135148415c44b476403.png)
are positive numbers. Prove that ![\[\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+24\cdot\dfrac{\sqrt[3]{abc}}{a+b+c}\geq 11.\]](http://latex.artofproblemsolving.com/1/3/3/133528c378ea6bb72d534414b1dfe3a84e46f777.png)
2. Let
, 
and 
are positives. Prove that ![\[\frac{(a+b)(b+c)(c+a)}{abc}\ge\frac{17(a+b+c)}{3\sqrt[3]{abc}}-9. \]](http://latex.artofproblemsolving.com/a/9/e/a9e476212a87aa50470f113d05c946ce3276e616.png)
3. Let
, and are positive numbers. Prove that![\[ \sqrt{\frac{b+c}{a}}+\sqrt{\frac{c+a}{b}}+\sqrt{\frac{a+b}{c}}\ge \sqrt{ 11\cdot\frac{a+b+c}{\sqrt[3]{abc}}-15 }.\]](http://latex.artofproblemsolving.com/e/4/3/e43ce425c843181d8f76e8975496a14c6d231213.png)
4. Let
, and are positive numbers. Prove that ![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge\sqrt{15\cdot\frac{a^2+b^2+c^2}{ab+bc+ca}-6}.\]](http://latex.artofproblemsolving.com/4/0/d/40db9f980fd659e134477be5690644f4810dbfcc.png)
5. For positives
, and prove that![\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{9abc}{a^2+b^2+c^2}\ge 2(a+b+c).\]](http://latex.artofproblemsolving.com/6/c/e/6ceb38df797ff34e4264361ef3fc5d837b7f45a4.png)
6. For positives
, and prove that ![\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq\frac{37(a^2+b^2+c^2)-19(ab+ac+bc)}{6(a+b+c)}.\]](http://latex.artofproblemsolving.com/a/b/1/ab17eb90ce1d96142dd1daa801135121f48af422.png)
7. For positives
, and prove that ![\[(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq9\left(\frac{a^2+b^2+c^2}{ab+ac+bc}\right)^{\frac{4}{7}}.\]](http://latex.artofproblemsolving.com/6/c/0/6c092d438e41b1b7c76fddc8fc0ff358dc0086ca.png)
8. Let
, and are non-negative numbers such that 
. Prove that![\[\sqrt a + \sqrt b + \sqrt c \ge \sqrt[7]{{9{{(ab + bc + ca)}^5}}}.\]](http://latex.artofproblemsolving.com/1/6/0/16067b3123264797bbb34135148415c44b476403.png)
This post has been edited 2 times. Last edited by Nguyenhuyen_AG, May 17, 2012, 4:05 PM
.![\[a\left( {{b}^{2}}+{{c}^{2}} \right)+b\left( {{c}^{2}}+{{a}^{2}} \right)+c\left( {{a}^{2}}+{{b}^{2}} \right)\ge 2\left( \sqrt{a^3}+\sqrt{b^3}+\sqrt{c^3}\right)\]](http://latex.artofproblemsolving.com/a/9/6/a9699a5749fe9df73e5c4096f5836309f33bfd19.png)
,
,
,
,
,
,
and the inequality becomes ![\[u^3 (u + v)^6 (2 u + v) (u^2 + u v + v^2) +
3 u^2 (u + v)^5 (8 u^4 + 15 u^3 v + 15 u^2 v^2 + 7 u v^3 + v^4) x +
3 u (u + v)^4 (44 u^5 + 99 u^4 v + 105 u^3 v^2 + 58 u^2 v^3 +
14 u v^4 + v^5) x^2 + (421 u^9 + 2385 u^8 v + 5940 u^7 v^2 +
8526 u^6 v^3 + 7746 u^5 v^4 + 4560 u^4 v^5 + 1690 u^3 v^6 +
360 u^2 v^7 + 36 u v^8 + v^9) x^3 +
3 (281 u^8 + 1442 u^7 v + 3198 u^6 v^2 + 3980 u^5 v^3 +
3048 u^4 v^4 + 1460 u^3 v^5 + 415 u^2 v^6 + 60 u v^7 +
3 v^8) x^4 +
6 (2 u + v) (91 u^6 + 378 u^5 v + 639 u^4 v^2 + 550 u^3 v^3 +
266 u^2 v^4 + 68 u v^5 + 6 v^6) x^5 + (916 u^6 + 3882 u^5 v +
6555 u^4 v^2 + 5506 u^3 v^3 + 2607 u^2 v^4 + 690 u v^5 +
76 v^6) x^6 +
6 (81 u^5 + 315 u^4 v + 440 u^3 v^2 + 264 u^2 v^3 + 84 u v^4 +
13 v^5) x^7 +
3 (52 u^4 + 185 u^3 v + 195 u^2 v^2 + 62 u v^3 +
10 v^4) x^8 + (29 u^3 + 84 u^2 v + 57 u v^2 + v^3) x^9 +
3 (u^2 + u v + v^2) x^{10} \geq 0\]](http://latex.artofproblemsolving.com/a/e/0/ae07ddc31ef8c9772643fcda2228f7cf97ffd296.png)
which is obvious.
,
,
,![\[u^6 (u + v)^3 (2 u + v) (u^2 + u v + v^2) +
3 u^5 (u + v)^2 (8 u^4 + 17 u^3 v + 18 u^2 v^2 + 10 u v^3 +
2 v^4) x +
3 u^4 (u + v) (44 u^5 + 121 u^4 v + 149 u^3 v^2 + 103 u^2 v^3 +
37 u v^4 + 5 v^5) x^2 + (421 u^9 + 1404 u^8 v + 2016 u^7 v^2 +
1638 u^6 v^3 + 816 u^5 v^4 + 276 u^4 v^5 + 94 u^3 v^6 +
36 u^2 v^7 + 9 u v^8 + v^9) x^3 +
3 (281 u^8 + 806 u^7 v + 972 u^6 v^2 + 662 u^5 v^3 + 318 u^4 v^4 +
158 u^3 v^5 + 79 u^2 v^6 + 24 u v^7 + 3 v^8) x^4 +
6 (2 u + v) (91 u^6 + 168 u^5 v + 114 u^4 v^2 + 46 u^3 v^3 +
35 u^2 v^4 + 26 u v^5 + 6 v^6) x^5 + (916 u^6 + 1614 u^5 v +
885 u^4 v^2 + 214 u^3 v^3 + 339 u^2 v^4 + 312 u v^5 +
76 v^6) x^6 +
6 (81 u^5 + 90 u^4 v - 10 u^3 v^2 - 24 u^2 v^3 + 21 u v^4 +
13 v^5) x^7 +
3 (52 u^4 + 23 u^3 v - 48 u^2 v^2 - 19 u v^3 +
10 v^4) x^8 + (29 u^3 + 3 u^2 v - 24 u v^2 + v^3) x^9 +
3 (u^2 + u v + v^2) x^10 \geq 0\]](http://latex.artofproblemsolving.com/7/2/8/72872739fe67c31284f549cd33741a0f77d7e5c4.png)
this is obvious for 
,
,
,
,
,
,![\[p^6 (2 p + q)^3 (3 p + q) (3 p^2 + 3 p q + q^2) +
3 p^5 (2 p + q)^2 (127 p^4 + 206 p^3 q + 132 p^2 q^2 + 38 p q^3 +
4 q^4) x +
3 p^4 (2 p + q) (2461 p^5 + 4690 p^4 q + 3646 p^3 q^2 +
1421 p^2 q^3 + 272 p q^4 + 20 q^5) x^2 + (86391 p^9 +
189438 p^8 q + 174396 p^7 q^2 + 86738 p^6 q^3 + 25452 p^5 q^4 +
4896 p^4 q^5 + 822 p^3 q^6 + 144 p^2 q^7 + 18 p q^8 + q^9) x^3 +
3 (112926 p^8 + 223756 p^7 q + 186663 p^6 q^2 + 86184 p^5 q^3 +
25388 p^4 q^4 + 5652 p^3 q^5 + 1068 p^2 q^6 + 144 p q^7 +
9 q^8) x^4 +
6 (155820 p^7 + 275278 p^6 q + 205530 p^5 q^2 + 87036 p^4 q^3 +
24647 p^3 q^4 + 5220 p^2 q^5 + 768 p q^6 +
54 q^7) x^5 + (1856556 p^6 + 2866152 p^5 q + 1868928 p^4 q^2 +
698398 p^3 q^3 + 173016 p^2 q^4 + 28332 p q^5 + 2253 q^6) x^6 +
12 (222276 p^5 + 291200 p^4 q + 159222 p^3 q^2 + 48967 p^2 q^3 +
9154 p q^4 + 828 q^5) x^7 +
3 (915900 p^4 + 975082 p^3 q + 418248 p^2 q^2 + 93392 p q^3 +
9659 q^4) x^8 + (1977633 p^3 + 1598388 p^2 q + 476544 p q^2 +
57536 q^3) x^9 + 3 (314566 p^2 + 170844 p q + 26457 q^2) x^{10} +
3 (89366 p + 24345 q) x^{11} + 34308 x^{12} \geq 0\]](http://latex.artofproblemsolving.com/a/8/a/a8a5f05b8e4979b4354654fa3402f2b4788b7bff.png)
which is obvious.
,
,
,![\[p^3 q^2 (3 p^7 + p^6 q + 30 p^5 q^2 + 78 p^4 q^3 + 76 p^3 q^4 +
36 p^2 q^5 + 9 p q^6 + q^7) +
3 p^2 q (3 p^8 + 3 p^7 q + 24 p^6 q^2 + 252 p^5 q^3 + 438 p^4 q^4 +
312 p^3 q^5 + 108 p^2 q^6 + 18 p q^7 + q^8) u +
3 p (3 p^9 + 16 p^8 q - 63 p^7 q^2 + 560 p^6 q^3 + 2497 p^5 q^4 +
2964 p^4 q^5 + 1511 p^3 q^6 + 360 p^2 q^7 + 36 p q^8 +
q^9) u^2 + (99 p^9 - 243 p^8 q - 432 p^7 q^2 + 16038 p^6 q^3 +
38736 p^5 q^4 + 31392 p^4 q^5 + 10974 p^3 q^6 + 1656 p^2 q^7 +
90 p q^8 + q^9) u^3 +
3 (180 p^8 - 570 p^7 q + 3405 p^6 q^2 + 26638 p^5 q^3 +
38568 p^4 q^4 + 20808 p^3 q^5 + 4653 p^2 q^6 + 396 p q^7 +
9 q^8) u^4 +
18 (159 p^7 + 243 p^6 q + 4392 p^5 q^2 + 12844 p^4 q^3 +
11451 p^3 q^4 + 3888 p^2 q^5 + 495 p q^6 +
18 q^7) u^5 + (15696 p^6 + 62802 p^5 q + 265326 p^4 q^2 +
402926 p^3 q^3 + 215436 p^2 q^4 + 41220 p q^5 + 2253 q^6) u^6 +
18 (3492 p^5 + 12895 p^4 q + 27568 p^3 q^2 + 23194 p^2 q^3 +
6776 p q^4 + 552 q^5) u^7 +
3 (53901 p^4 + 149027 p^3 q + 179181 p^2 q^2 + 79216 p q^3 +
9659 q^4) u^8 + (261717 p^3 + 489993 p^2 q + 317166 p q^2 +
57536 q^3) u^9 + 9 (28772 p^2 + 32317 p q + 8819 q^2) u^10 +
3 (47866 p + 24345 q) u^11 + 34308 u^12 \geq 0\]](http://latex.artofproblemsolving.com/d/e/d/ded9e54892639e26284ba70a74878a4a4eb9a475.png)
where each coefficient is easily verified to be non-negative, hence we are done!![\[3 u^3 (u + v)^3 (2 u + v) (u^2 + u v + v^2) +
9 u^2 (u + v)^2 (6 u^4 + 12 u^3 v + 12 u^2 v^2 + 6 u v^3 + v^4) x +
u (u + v) (2 u + v) (91 u^4 + 182 u^3 v + 155 u^2 v^2 + 64 u v^3 +
9 v^4) x^2 + (305 u^6 + 915 u^5 v + 1099 u^4 v^2 + 673 u^3 v^3 +
240 u^2 v^4 + 56 u v^5 + 6 v^6) x^3 + (2 u + v) (138 u^4 +
276 u^3 v + 160 u^2 v^2 + 22 u v^3 + 19 v^4) x^4 + (134 u^4 +
268 u^3 v + 111 u^2 v^2 - 23 u v^3 + 5 v^4) x^5 + (2 u +
v) (16 u^2 + 16 u v - 11 v^2) x^6 + 3 (u^2 + u v + v^2) x^7 \geq 0\]](http://latex.artofproblemsolving.com/1/c/3/1c3dbd4929f78781f51c8e305d722a4a6cc0ca37.png)
this is obvious for 
,![\[3 p^3 (2 p + q)^3 (3 p + q) (3 p^2 + 3 p q + q^2) +
9 p^2 (2 p + q)^2 (91 p^4 + 140 p^3 q + 84 p^2 q^2 + 22 p q^3 +
2 q^4) x +
p (2 p + q) (10719 p^5 + 18591 p^4 q + 12858 p^3 q^2 +
4301 p^2 q^3 + 667 p q^4 + 36 q^5) x^2 + (79776 p^6 +
153450 p^5 q + 119328 p^4 q^2 + 47053 p^3 q^3 + 9595 p^2 q^4 +
904 p q^5 + 27 q^6) x^3 + (186543 p^5 + 299784 p^4 q +
187830 p^3 q^2 + 56548 p^2 q^3 + 8020 p q^4 + 418 q^5) x^4 +
3 (94932 p^4 + 122103 p^3 q + 57639 p^2 q^2 + 11764 p q^3 +
875 q^4) x^5 +
3 (94773 p^3 + 91237 p^2 q + 28738 p q^2 + 2973 q^3) x^6 +
18 (9957 p^2 + 6360 p q + 998 q^2) x^7 + 9 (7202 p + 2283 q) x^8 +
10260 x^9 \geq 0\]](http://latex.artofproblemsolving.com/7/9/a/79a1f4de034e6dbec019134bf42562ed0f60386f.png)
which is obvious.
,![\[p^3 q^2 (3 p^4 - 11 p^3 q + 5 p^2 q^2 + 19 p q^3 + 6 q^4) +
3 p^2 q (3 p^5 - 6 p^4 q - 23 p^3 q^2 + 60 p^2 q^3 + 56 p q^4 +
9 q^5) u +
p (9 p^6 + 66 p^5 q - 99 p^4 q^2 + 454 p^3 q^3 + 1280 p^2 q^4 +
535 p q^5 + 45 q^6) u^2 +
3 (42 p^6 + 216 p^5 q + 546 p^4 q^2 + 1533 p^3 q^3 + 1255 p^2 q^4 +
256 p q^5 + 9 q^6) u^3 + (1062 p^5 + 5394 p^4 q + 12285 p^3 q^2 +
13873 p^2 q^3 + 5105 p q^4 + 418 q^5) u^4 +
3 (1860 p^4 + 7503 p^3 q + 10959 p^2 q^2 + 6074 p q^3 +
875 q^4) u^5 +
3 (5733 p^3 + 15889 p^2 q + 13178 p q^2 + 2973 q^3) u^6 +
18 (1669 p^2 + 2772 p q + 998 q^2) u^7 + 9 (3058 p + 2283 q) u^8 +
10260 u^9 \geq 0\]](http://latex.artofproblemsolving.com/5/f/0/5f01958ba3a7f7627119975d28035afa9ccdc221.png)
where each coefficient is easily verified to be non-negative.
,
,
,![\[a^4 (u^2 + u v + v^2) + u (u + v)^3 (2 u^2 + 2 u v + v^2) +
a^3 (3 u^3 + 9 u^2 v + 8 u v^2 + v^3) +
3 a^2 (2 u^4 + 7 u^3 v + 9 u^2 v^2 + 4 u v^3 + v^4) +
a (6 u^5 + 22 u^4 v + 33 u^3 v^2 + 23 u^2 v^3 + 8 u v^4 + v^5) \geq 0\]](http://latex.artofproblemsolving.com/e/0/1/e013be9977255fe112fe55f81d6fd91bd2062023.png)
which is obvious.
,
,
,![\[a^4 (u^2 + u v + v^2) + u^3 (u + v) (2 u^2 + 2 u v + v^2) +
a^3 (3 u^3 - u v^2 + v^3) + 3 a^2 (2 u^4 + u^3 v + u v^3 + v^4) +
a (6 u^5 + 8 u^4 v + 5 u^3 v^2 + 4 u^2 v^3 + 3 u v^4 + v^5) \geq 0\]](http://latex.artofproblemsolving.com/0/7/4/074d565cbf5c803b8ed41852bd3dbe157ae5f63d.png)
which is obvious.
.
.
![${ a,b,c>0\Rightarrow\frac{(a+b)(b+c)(c+a)}{abc}\ge\frac{17(a+b+c)}{3\sqrt[3]{abc}}-9} $](http://latex.artofproblemsolving.com/6/5/c/65cf923dee845590996affe5c2b590be7335e01e.png)
![$ \frac{a+b+c}{\sqrt[3]{abc}}=\sqrt[3]{a^2b^2c^2}(\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}\le $](http://latex.artofproblemsolving.com/3/1/5/3159475ad703b4ca08ce342d3772136de980213b.png)
that is true!
![$ a,b,c>0\Rightarrow\frac{(b+c)(c+a)(a+b)}{abc}\ge\frac{17(a+b+c)}{3\sqrt[3]{abc}}-9 $](http://latex.artofproblemsolving.com/2/0/8/208bdea2ac99dfcae83a317ab9bf2f053054506c.png)
;
;
is increasing,therefore
where
is negative and 
result 
;![$ a,b,c>0\Rightarrow\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+24\cdot\frac{\sqrt[3]{abc}}{a+b+c}\geq 11. $](http://latex.artofproblemsolving.com/e/c/7/ec7e7b05125c5222fc75860e642c0e628451495e.png)
where:
