19. Own proposed problems in CRUX Mathematicorum - Canada.

by Virgil Nicula, Apr 21, 2010, 1:35 AM

$1\blacktriangleright\ \boxed{\ \{a,b,c\}\subset\mathbb R\ \mathrm{and}\ a^2\ +\ b^2\ +\ c^2\ =\ 9\ \Longrightarrow\ 3\ \min\ \{\ a\ ,\ b\ ,\ c\ \}\ \le\ 1\ +\ abc\ }$ (3241/4/2007). See here my proof.


$2\blacktriangleright$ Let $A\subset\mathbb C^*$ so that $(\forall )\ z\in A\ ,\ \left|\ z\ +\ \frac 1z\ \right|\ \le\ 2$ . Prove that $(\forall )\ n\in\mathbb{N}^*\ ,\ \alpha^n\in A\ \Longrightarrow\ \alpha\in A$ (3242/4/2007). See here my proof.


$3\blacktriangleright$ Prove that $(\forall )$ positive $a\ne b\ ,\ (a-1)(b-1)>0\ \implies\ \boxed{\ a^b\ +\ b^a\ \ge\ 1\ +\ ab\ +\ (1-a)\ (1-b)\ \min\ \{\ 1\ ,\ ab\ \}\ }$ (3260/5/2007). See here my proof.


$4\blacktriangleright$ Let $\triangle\ ABC$ with $\left\|\ \begin{array}{c} M\in (BC)\ ,\ MB=MC\\\\ D\in (BC)\ ,\ \widehat {DAB}\equiv\widehat {DAC}\end{array}\ \right\|$ . Prove that $\boxed{\ A=90^{\circ}+m(\angle MAD)\ \Longleftrightarrow\ \frac bc=1-2\cos A\ }$(3264/6/2007).


$5\blacktriangleright$ Let a trapezoid $ABCD$ , $AB\parallel CD$ so that $DA=DC$ , $CA=CB$ . Denote $\left\{\begin{array}{c} m(\widehat{ABC})=x\\\\ m(\widehat{BDC})=y\\\\ m(\widehat{AED})=z\end{array}\right\|$ . Prove that $\left\{\begin{array}{c} y\ <\ 30^{\circ}\\\\ \tan y=\frac {2\tan x}{3+\tan^2x}\\\\ \tan z=\frac {2\sin x+\sin 3x}{2\cos x+\cos 3x}\end{array}\right\|$ (3265/6/2007).

$6\blacktriangleright$ Let two lines $k\ ,\ l$ (parallelly or concurrently) and two points $M\in k\ ,\ N\in l$ . For a mobile point $P$ so that $\delta_k(P)=\delta_l(P)$ denote

$\left\{\begin{array}{ccc} A\in k & ; & PA\perp k\\\\ B\in l & ; & PB\perp l\end{array}\right\|$ . Prove that $BM\perp PN\Longleftrightarrow AN\perp PM\Longleftrightarrow MN^2=AM^2+BN^2$ (3270/6/2007). See here some proofs.


$7\blacktriangleright\ \ |a+b|+|b+c|+|c+a|\ \le\ 2\ \Longleftrightarrow$ $|a|\le 1\ \wedge\ |b|\le 1\ \wedge\ |c|\le 1\ \wedge\ |a+b+c|\le 1$ (3271/6/2007).


$8\blacktriangleright$ Outside of $\triangle ABC$ construct isosceles triangles $\left\{\begin{array}{ccc} BMC & : & MB=MC\ ,\ m(\widehat {MBC})=m(\widehat {MCB})=\alpha\\\\ CNA & : & NC=NA\ ,\ m(\widehat {NAC})=m(\widehat {NCA})=\beta\\\\ APB & : & PA=PB\ ,\ m(\widehat {PAB})=m(\widehat {PBA})=\gamma\end{array}\right\|$

so that $\alpha +\beta +\gamma=90^{\circ}$ . Prove that the angles of $\triangle\ MNP$ have the values $\left\{\begin{array}{c} M=\beta +\gamma\\\\ N=\gamma+\alpha \\\\ P=\alpha +\beta\end{array}\right\|$ (3273/6/2007).


$9\blacktriangleright$ Let $\triangle ABC$ and $P$ for which $PC=PB$ si $PA=AB$ . Denote $x=m(\widehat {PBC})$ . Prove that $\sin (B-C)=2\sin C\cos (B+2x)$ (3278/7/2007).


$10\blacktriangleright$ Let $\triangle ABC$ with the circumcircle $w=C(O,R)$ . Let $[AT]$ be a diameter of $w$ and $E\in AB\ ,\ F\in AC$ for which

$\left\{\begin{array}{c} O\in EF\ ;\ OE=OF\\\\ P\in EF\ ;\ TP\perp EF\end{array}\right\|$ . Prove that $BC$ , $EF\ ,$ the tangent line at $T$ to $w$ are concurrently and $\widehat {TPB}\equiv\widehat {TPC}$ (3280/7/2007).


$11\blacktriangleright$ Let $x$ , $y$ , $z$ so that $xy+yz+zx+xyz=4$ . Prove that exists $\triangle ABC$ for which $\left\{\begin{array}{c} a=(y+2)(z+2)\\\\ b=(z+2)(x+2)\\\\ c=(x+2)(y+2)\end{array}\right\|$

and show that $\boxed{\ 1\ +\ x\ +\ y\ +\ z\ \le\ xyz\ +\ \frac 1x\ +\ \frac 1y\ +\ \frac 1z\ }$ (3287/7/2007).


$12\blacktriangleright$ Let $\triangle ABC$ with the circumcircle $C(O,R)$ , the incircle $C(I,r)$ and $\left\{\begin{array}{c} M\in OI\cap AB\\\\ N\in OI\cap AC\end{array}\right\|$ . Prove that

$BMNC$ is cyclically $\Longleftrightarrow\ h_a=R+r$ and in this case $\frac {1}{MN}=\frac 1a+\frac {1}{b+c}$ (3279/8/2007).


$13\blacktriangleright$ Let $\triangle ABC$ and for its interior $D$ so that $\left\{\begin{array}{c} \widehat {ACD}\equiv\widehat {BAD}\\\\ \widehat {CAD}\equiv\widehat {ABD}\end{array}\right\|$ let $\left\{\begin{array}{ccccc} E\in AB & , & B\in (AE) & , & AB=BE\\\\ F\in CA & , & A\in (CF) & , & AC=CF\end{array}\right\|$ . Prove that $ADEF$ is cyclic (3289/8/2007).


$14\blacktriangleright$ Let a trapezoid $ABCD$ , $a=BC\ \parallel\ AD=b$ . Denote $\left\{\begin{array}{c} M\in (CD)\ ,\ MC=MD\\\\ P\in (AM)\ ,\ PA=PM\\\\ Q\in (BM)\ ,\ QB=QM\end{array}\right\|$ and $N\in CQ\cap DP$ .

Prove that $N$ belongs to the interior of $\triangle\ ABM\ \Longleftrightarrow\ \frac 13\ \le\ \frac ab\ \le\ 3$ (3290/8/2007).


$15\blacktriangleright$ Ascertain all points $P$ from the plane of $A$-isosceles $\triangle ABC$ so that for any line $d\ ,\ P\in d$ the sum $\delta^2_d(A)+\delta^2_d(B)+\delta^2_d(C)$ is constant (3291/8/2007).


$\boxed{\ \mathrm{Let}\ A\subset\mathbb C^*\ \mathrm{so\ that}\ (\forall )\ z\in A\ ,\ \left|\ z\ +\ \frac 1z\ \right|\ \le\ 2\ .\ \mathrm{Prove\ that}\ (\forall )\ n\in\mathbb{N}^*\ ,\ \alpha^n\in A\ \Longrightarrow\ \alpha\in A\ }$
This post has been edited 78 times. Last edited by Virgil Nicula, Sep 30, 2017, 5:41 PM

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Virgil Nicula
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28. A fost odată ...
27.Problem of Darij Grinberg (I - centroid of triangle DEF).
26. Outside of a triangle.
25. Geometric equations.
24. A fine and cool inequalities.
23. A very nice exponential inequality (own - from CRUX).
22. Some interesting limits (sequence/function).
21 bis. Some problems with complex numbers.
21. Probleme de geometrie (Yetty & I).
20. O ineg.cu suma de AG/AI .
19. Own proposed problems in CRUX Mathematicorum - Canada.
18. O problema a lui Toshio Seimiya.
17. O inegalitate "tare" intr-un triunghi.
16. Length of the (interior) angle bisector (10 proofs).
15. Inequalities stronger than Panaitopol's.
14. Opinii si comentarii relativ la inv. rom.
13. Inegalitatea de la Sorin Borodi.
12. Inegalitatea I. Maftei & S. Radulescu (2).
11. Inegalitatea I. Maftei & S. Radulescu (1).
10. Walker's inequality and its extension.
9. Inegalitatea HO+HA+HB+HC >= 3R .
8. The first problem Shortlist ONM 2010 Romania.
7. Some interesting "slicing" problems.
6. Interesting problems from JBMO.
5. Theoretical preliminary for inequalities.
4. Remarkable geometrical inequalities (2).
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    by ryanbear, May 9, 2023, 6:10 AM

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    by OlympusHero, Sep 14, 2022, 4:44 AM

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    by tigerzhang, Aug 1, 2021, 12:02 AM

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    by GoogleNebula, Apr 14, 2021, 5:25 AM

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    by samrocksnature, Apr 14, 2021, 5:16 AM

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    by the_mathmagician, Jan 17, 2021, 7:43 PM

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    by OlympusHero, Dec 30, 2020, 6:08 PM

  • long blog

    by MrMustache, Nov 11, 2020, 4:52 PM

  • 372554 views!

    by mrmath0720, Sep 28, 2020, 1:11 AM

  • wow... i am lost.

    369302 views!

    -piphi

    by piphi, Jun 10, 2020, 11:44 PM

  • That was a lot! But, really good solutions and format! Nice blog!!!! :)

    by CSPAL, May 27, 2020, 4:17 PM

  • impressive :D
    awesome. 358,000 visits?????

    by OlympusHero, May 14, 2020, 8:43 PM

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  • Joined: Jun 22, 2005
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  • Total entries: 456
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