Inspired by IMO 1984

by sqing, Mar 26, 2025, 3:49 AM

Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that
$$a+b+\frac{9}{5}c\geq\frac{9}{8}$$$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+ ab +18abc\geq\frac{343}{324} $. Prove that
$$a+b+\frac{6}{5}c\geq\frac{7\sqrt 7}{18}$$$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$
This post has been edited 1 time. Last edited by sqing, 3 hours ago
inequalities proposed
algebra
L

integral points

by jhz, Mar 26, 2025, 1:14 AM

Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions:
$(1)$ $|x_i|,|y_i|\le 10^{10} $ for all $1\le i \le 10$
$(2)$ Define the set \[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]then \(|S| = 1024\)，and any rectangular strip of width 1 covers at most two points of S.
This post has been edited 1 time. Last edited by jhz, 6 hours ago
combinatorics
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equal angles

by jhz, Mar 26, 2025, 12:56 AM

In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
geometry
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Additive Combinatorics!

by EthanWYX2009, Mar 25, 2025, 12:49 AM

Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\]\[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\]\[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\]Show that \( |A|^2 \leq |B| \cdot |C| \).
combinatorics
Additive combinatorics
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2 degree polynomial

by PrimeSol, Mar 24, 2025, 6:13 AM

Let $P_{1}(x)= x^2 +b_{1}x +c_{1}, ... , P_{n}(x)=x^2+ b_{n}x+c_{n}$, $P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$ $\forall i,j ,1 \leq i<j \leq n : P_{i}(x) \ne P_{j}(x)$.
$\forall i,j, 1\leq i<j \leq n : Q_{i,j}(x)= P_{i}(x) + P_{j}(x)$ polynomial with only one root.
$max(n)=?$
This post has been edited 8 times. Last edited by PrimeSol, Mar 24, 2025, 6:34 AM
algebra
polynomial
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D1010 : How it is possible ?

by Dattier, Mar 10, 2025, 10:49 AM

Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
This post has been edited 6 times. Last edited by Dattier, Mar 16, 2025, 10:10 AM
number theory
Computer solutions
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7 triangles in a square

by gghx, Oct 12, 2024, 11:29 AM

Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.
combinatorics
area
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Smallest value of |253^m - 40^n|

by MS_Kekas, Jan 28, 2024, 9:35 PM

Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
number theory
Diophantine equation
L

Circumcenter lies on line parallel to BC

by math_pi_rate, Aug 20, 2018, 12:14 PM

Here is a problem I made a few days ago (which I'll later be using as a lemma).
LEMMA Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Let $D$ be the midpoint of arc $\overarc{BC}$ (not containing $A$), and let $K$ be the antipode of $D$ in $\omega$. Let $AK \cap CD = T$. Finally let $O$ be the circumcenter of $\triangle ABT$. Then $OD$ is tangent to $\omega$.

Solution 1 (My solution, along with some motivation)

Solution 2 (Solution by TheDarkPrince)

Anyway, here's a problem that uses this lemma.
Problem (Polish MO 2018 P5): Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.

Solution
This post has been edited 6 times. Last edited by math_pi_rate, Sep 27, 2018, 12:58 PM
geometry
Circumcenter
external angle bisector
arc midpoint

n-variable inequality

by ABCDE, Jul 7, 2016, 7:34 PM

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\]for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
algebra
IMO Shortlist
Sequence
Inequality
induction
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Flee Jumping on Number Line

by utkarshgupta, Dec 11, 2015, 3:59 PM

An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
This post has been edited 2 times. Last edited by djmathman, Apr 15, 2016, 5:59 PM
combinatorics
number theory
All Naturals
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Geo Geo everywhere, nor a point to see.

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math_pi_rate
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  • duh i found this masterpiece after the owner went inactive noooooooooooo :sadge: :(

    by Project_Donkey_into_M4, Nov 23, 2021, 2:56 PM

  • Hi everyone! Sorry but I doubt if I'll be reviving this anytime soon :oops:

    I might post something if I find anything interesting, but they'll mostly be short posts, unlike the previous ones!

    by math_pi_rate, Nov 7, 2020, 7:39 PM

  • @below be patient. He must be busy with some other work. :)

    by amar_04, Oct 22, 2020, 10:23 AM

  • Advanced is over now, please revive this please!

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  • REVIVE please

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  • re-vi-ve

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  • Try this: Prove that a two colored cow has an odd number of Agis Phesis

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