Romanian National Olympiad 1997 - Grade 9 - Problem 4

by Filipjack, Apr 6, 2025, 8:34 PM

Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$ $B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$ If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.

floor function

algebra

equations

Romanian National Olympiad 1997 - Grade 9 - Problem 2

by Filipjack, Apr 6, 2025, 8:24 PM

Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$

function

algebra

Giving n books when you have n1 + 1(2n+1) books

by Miquel-point, Apr 6, 2025, 8:05 PM

At a maths contest $n$ books are given as prizes to $n$ students (each students gets one book). In how many ways can the organisers give these prizes if they have $n$ copies of one book and $2n+1$ other books each in one copy?

combinatorics

Finding signs in a nice inequality of L. Panaitopol

by Miquel-point, Apr 6, 2025, 8:00 PM

Consider $x_1,\ldots,x_n>0$ . Show that there exists $a_1,a_2,\ldots,a_n\in \{-1,1\}$ such that
$a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.$
Laurențiu Panaitopol

algebra

inequalities

Right tetrahedron of fixed volume and min perimeter

by Miquel-point, Apr 6, 2025, 7:57 PM

Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.

s(n) and s(n+1) divisible by m

by Miquel-point, Apr 6, 2025, 6:26 PM

Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$ , where $s(x)$ is the sum of digits of $x$ .

Dorel Miheț

number theory

Pythagorean new journey

by XAN4, Apr 6, 2025, 3:41 AM

The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$ , if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.

number theory

number theory unsolved

interesting ineq

by nikiiiita, Jan 29, 2025, 10:27 AM

Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$ . Prove that:
$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$

This post has been edited 1 time. Last edited by nikiiiita, Jan 29, 2025, 10:29 AM

inequalities

Convex and concave functions in Real numbers -- Basic 1

by adityaguharoy, Mar 1, 2018, 1:44 PM

Convex functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a convex function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \le t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly convex on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Concave functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a concave function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \ge t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly concave on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Quick exercises

Let us celebrate

This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM

combinatorics and number theory beautiful problem

by Medjl, Feb 1, 2018, 3:16 PM

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$

number theory

Path within S which does not meet itself

by orl, Nov 11, 2005, 9:35 PM

Let $S$ be a square with sides length $100$ . Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$ . Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$ . Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$ .

geometry

combinatorial geometry

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2025 shout!
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2024 shout
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hullo :<
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hello $\text{[math]}$
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Halo thear
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hi!!
just found this and I can't wait to read more!
so happy to have found this blog!
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still waiting for the mathlinks camp lol
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hello
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Right below the shout box it says how many it has.
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Nice blog!
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by bobert1, Jun 30, 2016, 9:29 PM
yeah sure we must and will discuss real numbers.
by adityaguharoy, Jun 20, 2016, 4:11 AM
Why not real numbers
by skipiano, Jun 17, 2016, 2:35 PM
very interesting topic !! ?? !!
by adityaguharoy, Jun 12, 2016, 6:52 AM
Hello!!!!!!! Very interesting topic, here!
by MinionLA, Jun 9, 2016, 11:43 PM
Should it not start by the discussion of Set Theory.
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Darn I missed it
by skipiano, Jun 9, 2016, 4:52 AM
first shout >
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117 shouts

An Introduction to the Theory of Mathematics

by Filipjack, Apr 6, 2025, 8:34 PM

by Filipjack, Apr 6, 2025, 8:24 PM

by Miquel-point, Apr 6, 2025, 8:05 PM

by Miquel-point, Apr 6, 2025, 8:00 PM

by Miquel-point, Apr 6, 2025, 7:57 PM

by Miquel-point, Apr 6, 2025, 6:26 PM

by XAN4, Apr 6, 2025, 3:41 AM

by nikiiiita, Jan 29, 2025, 10:27 AM

by adityaguharoy, Mar 1, 2018, 1:44 PM

by Medjl, Feb 1, 2018, 3:16 PM

by orl, Nov 11, 2005, 9:35 PM