Search found 9 matches
- Fri Jul 02, 2021 5:21 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 6
- Replies: 1
- Views: 5052
Re: BdMO National 2021 Higher Secondary Problem 6
\(ABC\) হলো একটা সূক্ষ্মকোণী ত্রিভুজ। \(\angle BAC\)-এর বহির্দ্বিখণ্ডক \(BC\) রেখাকে \(N\) বিন্দুতে ছেদ করে। \(BC\)-এর মধ্যবিন্দু হলো \(M\)। \(P\) আর \(Q\) হলো \(AN\) রেখার ওপর এমন দুটো বিন্দু যেন \(\angle PMN =\angle MQN=90^\circ\)। যদি \(PN=5\) আর \(BC=3\) হয়, তাহলে \(QA\)-এর দৈর্ঘ্যকে \(\frac{a...
- Fri Jul 02, 2021 2:41 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 10
- Replies: 1
- Views: 4803
Re: BdMO National 2021 Higher Secondary Problem 10
\(A_1A_2A_3A_4A_5A_6A_7A_8\) একটা সুষম অষ্টভুজ। \(P\) এই অষ্টভুজের মধ্যে এমন একটা বিন্দু যেন \(P\) বিন্দু থেকে \(A_1A_2\), \(A_2A_3\) আর \(A_3A_4\)-এর দূরত্ব যথাক্রমে \(24\), \(26\) আর \(27\)। \(A_1A_2\)-এর দৈর্ঘ্যকে \(a\sqrt{b}-c\) আকারে লেখা যায় যেখানে \(a\), \(b\) আর \(c\) ধনাত্মক পূর্ণসংখ্যা এ...
- Thu Jul 01, 2021 8:34 pm
- Forum: Algebra
- Topic: Inequality Marathon
- Replies: 9
- Views: 18081
Problem 04
Let $a$,$b$,$c$ be positive real numbers such that $a+b+c=3$. Prove that
$a^2+b^2+c^2\geq\frac{2+a}{2+b}+\frac{2+b}{2+c}+\frac{2+c}{2+a}$
$a^2+b^2+c^2\geq\frac{2+a}{2+b}+\frac{2+b}{2+c}+\frac{2+c}{2+a}$
- Thu Jul 01, 2021 8:20 pm
- Forum: Algebra
- Topic: Inequality Marathon
- Replies: 9
- Views: 18081
Solution of Problem 03
Let $a_1,a_2,a_3,\cdots,a_n$ be positive real numbers where $n\geq2, n\in\mathbb{N}$. Let $s=a_1+a_2+a_3+\cdots+a_n$. Prove that \[\frac{a_1}{s-a_1}+\frac{a_2}{s-a_2}+\frac{a_3}{s-a_3}+\cdots+\frac{a_n}{s-a_n}\geq\frac{n}{n-1}\] $Solution$ : $WLOG$, Let us consider, ${a_1}\geq{a_2}\geq{a_3}\geq\cdo...
- Wed May 12, 2021 12:52 am
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
- Replies: 6
- Views: 8569
Re: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
Let $\triangle ABC$ be a triangle inscribed in a circle $\omega$. $D,E$ are two points on the arc $BC$ of $\omega$ not containing $A$. Points $F,G$ lie on $BC$ such that \[\angle BAF = \angle CAD, \angle BAG = \angle CAE\] Prove that the two lines $DG$ and $EF$ meet on $\omega$. My solution is a bi...
- Tue May 11, 2021 2:43 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 7
- Replies: 2
- Views: 4973
Re: BdMO National 2021 Higher Secondary Problem 7
একটা বাইনারি স্ট্রিং হলো এমন একটা শব্দ যার মধ্যে খালি \(0\) আর \(1\) আছে। কোনো বাইনারি স্ট্রিংয়ে একটা \(1\)-রান হলো এমন একটা সাবস্ট্রিং যেটাতে খালি \(1\) আছে এবং যেটাকে আর ডানে বা বামে বড় করা যায় না। কোনো একটা ধনাত্মক পূর্ণসংখ্যা \(n\)-এর জন্য \(B(n)\) হলো \(n\)-কে বাইনারিতে লিখলে এতে যতগুলো \(1...
- Sun May 09, 2021 5:20 pm
- Forum: National Math Camp
- Topic: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
- Replies: 6
- Views: 8569
Re: Problem - 01 - National Math Camp 2021 Geometry Test - "The intersection point lies on the circumcircle"
Let $\triangle ABC$ be a triangle inscribed in a circle $\omega$. $D,E$ are two points on the arc $BC$ of $\omega$ not containing $A$. Points $F,G$ lie on $BC$ such that \[\angle BAF = \angle CAD, \angle BAG = \angle CAE\] Prove that the two lines $DG$ and $EF$ meet on $\omega$. $\textbf{Solution}$...
- Sun May 09, 2021 3:21 pm
- Forum: National Math Camp
- Topic: National Math Camp Geometry Exam Problem-2
- Replies: 1
- Views: 5862
National Math Camp Geometry Exam Problem-2
$Problem$ $2$. $ABC$ is a triangle where $∠BAC = 90◦$. A line through the midpoint $D$ of $BC$ meets $AB$ at $X$ and $AC$ at $Y$, where $X$ and $Y$ are not on the same side of $BC$. The point $P$ is taken on $XY$ such that $PD$ and $XY$ have the same midpoint $M$. The perpendicular from $P$ to $BC$ ...
- Sun May 09, 2021 3:13 pm
- Forum: National Math Camp
- Topic: National Math Camp Geometry Exam Problem-3
- Replies: 2
- Views: 10324
National Math Camp Geometry Exam Problem-3
$Problem$ $3$.
Let $ABC$ be a triangle with $BC$ being the longest side. Let $O$ be the circumcenter
of $ABC$. $P$ is an arbitrary point on $BC$. The perpendicular bisector of $BP$ meet $AB$ at $Q$ and the
perpendicular bisector of $PC$ meet $AC$ at $R$. Prove that $AQOR$ is cyclic.
Let $ABC$ be a triangle with $BC$ being the longest side. Let $O$ be the circumcenter
of $ABC$. $P$ is an arbitrary point on $BC$. The perpendicular bisector of $BP$ meet $AB$ at $Q$ and the
perpendicular bisector of $PC$ meet $AC$ at $R$. Prove that $AQOR$ is cyclic.