Found problems: 6530
1997 Vietnam Team Selection Test, 1
Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying
\[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
\]
and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
1987 Dutch Mathematical Olympiad, 2
For $x >0$ , prove that $$\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}$$
and for all $n \ge 2$ prove that $$1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2$$
2013 Miklós Schweitzer, 1
Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers:
\[|A+qA|\ge (q+1)|A|-C_q.\]
[i]Proposed by Antal Balog.[/i]
2003 Junior Tuymaada Olympiad, 4
The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$
2005 France Team Selection Test, 6
Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$.
Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.
the 12th XMO, Problem 5
Let $a,b,c\in\mathbb R_+$ satisfy that
$$\sqrt{(1+a)(1+b)(1+c)}=\sqrt{(ab-a-b+1)(1+c)}+\sqrt{(bc-b-c+1)(1+a)}+\sqrt{(ca-c-a+1)(1+b)}.$$
Find the value range of $a+b+c.$
2019 IFYM, Sozopol, 4
On a competition called [i]"Mathematical duels"[/i] students were given $n$ problems and each student solved exactly 3 of them. For each two students there is at most one problem that is solved from both of them. Prove that, if $s\in \mathbb{N}$ is a number for which $s^2-s+1<2n$, then there are $s$ problems among the $n$, no three of which solved by one student.
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}$
1996 APMO, 5
Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Prove that
\[ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c} \]
and determine when equality occurs.
2013 IFYM, Sozopol, 2
Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true:
$(a^2+b^2) cos(\alpha -\beta )\leq 2ab$.
1998 Romania National Olympiad, 1
Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .
2024 Junior Balkan Team Selection Tests - Moldova, 1
Let $a,b,c,x,y,z$ be positive real numbers, such that $a+b+c=xyz=1$ Prove that:
$$
\frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1
$$
When does equality hold?
2020 DMO Stage 1, 1.
[b]Q[/b] Let $p,q,r$ be non negative reals such that $pqr=1$. Find the maximum value for the expression
$$\sum_{cyc} p[r^{4}+q^{4}-p^{4}-p]$$
[i]Proposed by Aritra12[/i]
2011 N.N. Mihăileanu Individual, 3
Prove the inequalities $ 0<n\left( \sqrt[n]{2} -1 \right) -\left( \frac{1}{n+1} +\frac{1}{n+2} +\cdots +\frac{1}{n+n}\right) <\frac{1}{2n} , $ where $ n\ge 2. $
[i]Marius Cavachi[/i]
1965 Polish MO Finals, 1
Prove the theorem:
the lengths $ a$, $ b $, $ c $ of the sides of a triangle and the arc measures $ \alpha $, $ \beta $, $ \gamma $of its opposite angles satisfy the inequalities $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$
2001 India IMO Training Camp, 1
Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+
y+z)$.
2005 Alexandru Myller, 1
Let $ x,y,z $ be numbers distinct from $ -1 $ that verify the equation
$$ \frac{1}{1+a} +\frac{1}{1+b} +\frac{1}{1+c} =\frac{3}{2} . $$
Prove that if $ abc=1, $ then $ a $ or $ b $ or $ c $ is equal to $ 1. $
1990 Poland - Second Round, 5
There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.
1988 IMO Longlists, 92
Let $p \geq 2$ be a natural number. Prove that there exist an integer $n_0$ such that \[ \sum^{n_0}_{i=1} \frac{1}{i \cdot \sqrt[p]{i + 1}} > p. \]
PEN O Problems, 29
Let $A$ be a set of $N$ residues $\pmod{N^2}$. Prove that there exists a set $B$ of $N$ residues $\pmod{N^2}$ such that the set $A+B=\{a+b \vert a \in A, b \in B \}$ contains at least half of all the residues $\pmod{N^2}$.
2014 AMC 12/AHSME, 20
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
2019 Junior Balkan Team Selection Tests - Romania, 4
Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$).
Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$
.
2002 Singapore MO Open, 3
Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$
where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer
2019 Polish Junior MO First Round, 3
The integers $a, b, c$ are not $0$ such that $\frac{a}{b + c^2}=\frac{a + c^2}{b}$. Prove that $a + b + c \le 0$.
1995 Austrian-Polish Competition, 8
Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$.
(a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$.
(b) Find a smaller $d$ (the smaller, the better) with the same property.