3 var with parameter (on GMA 575)

by mihaig, Jul 28, 2025, 7:34 PM

Find the least constant $K$ such that
$$a+b+c+K\cdot\frac{\left(a-c\right)^2}{a+c}\geq\sqrt{3\left(a^2+b^2+c^2\right)}$$for all $a\ge b\ge c\ge0,$ with $a>0.$

Old or new triangle

by mihaig, Jul 28, 2025, 3:18 PM

Let $\Delta ABC$ be with no obtuse angles.
Prove
$$4\left(\cos A+\cos B+\cos C\right)^2+5\left(\sum \cos ^2A-\sum \cos A\cos B\right)\geq9.$$

One of the Craziest Problem I've ever seen (see the proposers)

by EthanWYX2009, Jul 17, 2025, 1:56 PM

For a positive integer \( n \), let \( f(n) \) be the minimal positive integer, such that for any \( n \) positive integers $x_1$, $x_2$, $\cdots$, $x_n$, \(\nu_2 \left(\sum_{i \in I} x_i\right)\) takes at most \( f(n) \) distinct integer values as \( I \) ranges over all non-empty subsets of \(\{1, 2, \cdots, n\}\).

Determine the value of \(\lim\limits_{n \to \infty} \dfrac{f(n)}{n \log_2 n}\).

Proposed by Zhenyu Dong from Hangzhou Xuejun High School, Chunji Wang from Shanghai High School, and Cheng Jiang from Tsinghua University
This post has been edited 1 time. Last edited by EthanWYX2009, Jul 17, 2025, 1:56 PM

Midpoint of arc black magic

by MarkBcc168, Jul 16, 2025, 3:01 AM

Let \(ABC\) be a triangle with incenter \(I\) such that \(AB<AC<BC\). The second intersections of \(AI\), \(BI\), and \(CI\) with the circumcircle of triangle \(ABC\) are \(M_{A}\), \(M_{B}\), and \(M_{C}\), respectively. Lines \(AI\) and \(BC\) intersect at \(D\) and lines \(BM_{C}\) and \(CM_{B}\) intersect at \(X\). Suppose the circumcircle of triangles \(XM_{B}M_{C}\) and \(XBC\) intersect again at \(S\neq X\). Lines \(BX\) and \(CX\) intersect the circumcircle of triangle \(SXM_{A}\) again at \(P\neq X\) and \(Q\neq X\), respectively.

Prove that the circumcenter of triangle \(SID\) lies on \(PQ\).

Proposed by Thailand

Complex number inequalities

by jhz, Mar 26, 2025, 12:28 AM

Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]
This post has been edited 1 time. Last edited by jhz, Mar 26, 2025, 12:28 AM

Parallel lines in two-circle configuration

by Tintarn, Apr 4, 2024, 6:43 PM

Let $ABC$ be an acute triangle, $\omega$ its circumcircle and $O$ its circumcenter. The altitude from $A$ intersects $\omega$ in a point $D \ne A$ and the segment $AC$ intersects the circumcircle of $OCD$ in a point $E \ne C$. Finally, let $M$ be the midpoint of $BE$. Show that $DE$ is parallel to $OM$.

Nice sequence bound

by VicKmath7, Feb 13, 2024, 11:21 AM

Let $N$ be a positive integer. The sequence $x_1, x_2, \ldots$ of non-negative reals is defined by $$x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}$$for all positive integers $n>N$. Show that there exists a constant $c>0$, such that $x_n \leq \frac{n} {2}+c$ for all positive integers $n$.

402 divides at least one of n^2-1, n^3-1, n^4-1

by Demetres, Feb 21, 2022, 4:12 PM

Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[n^2-1, n^3-1, n^4-1\]

Strange Conditional Sequence

by MarkBcc168, Jun 11, 2019, 12:18 AM

Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.
This post has been edited 1 time. Last edited by MarkBcc168, Jun 12, 2019, 2:42 PM

complex numbers inequality

by LeXuS, Mar 8, 2019, 9:07 AM

Let $x$, $y$, $z$ be 3 complex numbers with $|x|=|y|=|z|=1$. Prove that $$3 \leq |x+y-z|+|z+x-y|+|y+z-x| \leq 6$$

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