Found problems: 6530
2015 AMC 12/AHSME, 10
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2008 Harvard-MIT Mathematics Tournament, 8
Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.
2003 Mediterranean Mathematics Olympiad, 3
Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality
\[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]
2013 Canadian Mathematical Olympiad Qualification Repechage, 6
Let $x, y, z$ be real numbers that are greater than or equal to $0$ and less than or equal to $\frac{1}{2}$
[list]
[*] (a) Determine the minimum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this minimum is obtained.
[*] (b) Determine the maximum possible value of \[x+y+z-xy-yz-zx\] and determine all triples $(x,y,z)$ for which this maximum is obtained.[/list]
Kyiv City MO Seniors 2003+ geometry, 2003.11.3
Let $x_1, x_2, x_3, x_4$ be the distances from an arbitrary point inside the tetrahedron to the planes of its faces, and let $h_1, h_2, h_3, h_4$ be the corresponding heights of the tetrahedron. Prove that $$\sqrt{h_1+h_2+h_3+h_4} \ge \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}$$
(Dmitry Nomirovsky)
2021 Vietnam TST, 4
Let $a,b,c$ are non-negative numbers such that
$$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$
then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$
2003 China Team Selection Test, 1
Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.
2011 Akdeniz University MO, 3
For all $x \geq 2$, $y \geq 2$ real numbers, prove that
$$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$
1992 Baltic Way, 13
Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality
\[ \sum_{i\equal{}1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i\equal{}1}^n(x_i\plus{}y_i)^2}
\] holds.
Kvant 2021, M2635
In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$.
[i]Porposed by O. Titov[/i]
2001 Junior Balkan Team Selection Tests - Moldova, 6
Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero.
Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”.
a) Show that the sentence $(P)$ is true for $k = 2$.
b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$
MathLinks Contest 2nd, 1.1
Let $x, y, z$ be positive numbers such that $xyz \le 2$ and $\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k$, for some real $k \ge 2$. Find all values of $k$ such that the conditions above imply that there exist a triangle having the side-lengths $x, y, z$.
MathLinks Contest 4th, 4.1
Let $N_0$ be the set of all non-negative integers and let $f : N_0 \times N_0 \to [0, +\infty)$ be a function such that $f(a, b) = f(b, a)$ and $$f(a, b) = f(a + 1, b) + f(a, b + 1),$$ for all $a, b \in N_0$. Denote by $x_n = f(n, 0)$ for all $n \in N_0$.
Prove that for all $n \in N_0$ the following inequality takes place $$2^n x_n \ge x_0.$$
1961 Putnam, B1
Let $a_1 , a_2 , a_3 ,\ldots$ be a sequence of positive real numbers, define $s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}$ and $r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.$ Given that $\lim_{n\to \infty} s_n $ and $\lim_{n\to \infty} r_n $ exist, prove that the product of these limits is not less than $1.$
1994 Hungary-Israel Binational, 2
Let $ a_1$, $ \ldots$, $ a_k$, $ a_{k\plus{}1}$, $ \ldots$, $ a_n$ be $ n$ positive numbers ($ k<n$). Suppose that the values of $ a_{k\plus{}1}$, $ a_{k\plus{}2}$, $ \ldots$, $ a_n$ are fixed. Choose the values of $ a_1$, $ a_2$, $ \ldots$, $ a_k$ that minimize the sum $ \sum_{i, j, i\neq j}\frac{a_i}{a_j}$
2013 Vietnam Team Selection Test, 4
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
2017 Serbia National Math Olympiad, 1
Prove that for positive real numbers $a,b,c$ such that $a+b+c=1$,
$$a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.$$
2019 Latvia Baltic Way TST, 13
Let $s(k)$ denotes sum of digits of positive integer $k$. Prove that there are infinitely many positive integers $n$, which are not divisible by $10$ and satisfies:
$$s(n^2) < s(n) - 5$$
2014 Singapore MO Open, 5
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime.
1996 IMO Shortlist, 7
Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
1984 Swedish Mathematical Competition, 3
Prove that if $a,b$ are positive numbers, then
$$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$
1972 Kurschak Competition, 1
A triangle has side lengths $a, b, c$. Prove that
$$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$
1993 Poland - First Round, 10
Given positive real numbers $p,q$ with $p+q=1$. Prove that for all positive integers $m,n$ the following inequality holds
$(1-p^m)^n+(1-q^n)^m \geq 1$.
2019 Saudi Arabia JBMO TST, 4
Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $a, b, c, d \ge 0$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Prove that $$\frac{a + b + c + d}{2} \ge 1 + \sqrt{abcd}$$ When does the equality hold?
Leonard Giugiuc and Valmir B. Krasniqi