Found problems: 85335
2021 Vietnam TST, 3
Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$.
a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line.
b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.
1987 All Soviet Union Mathematical Olympiad, 441
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$
2017 Pakistan TST, Problem 3
Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$
$f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$
2018 Poland - Second Round, 4
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Circle of diameter $BC$ is tangent to line $AD$. Prove, that circle of diameter $AD$ is tangent to line $BC$.
IV Soros Olympiad 1997 - 98 (Russia), 10.5
At the base of the triangular pyramid $ABCD$ lies a regular triangle $ABC$ such that $AD = BC$. All plane angles at vertex $B$ are equal to each other. What might these angles be equal to?
2004 Federal Math Competition of S&M, 3
If $a,b,c$ are positive numbers such that $abc = 1$, prove the inequality
$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} \geq \sqrt{2}$
2005 South africa National Olympiad, 6
Consider the increasing sequence $1,2,4,5,7,9,10,12,14,16,17,19,\dots$ of positive integers, obtained by concatenating alternating blocks $\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots$ of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the $n$-th element of the sequence is given by \[2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.\]
(Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)
2021 USA TSTST, 3
Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$.
[i]Merlijn Staps[/i]
2009 India National Olympiad, 4
All the points in the plane are colored using three colors.Prove that there exists a triangle with vertices having the same color such that [i]either[/i] it is isosceles [i]or[/i] its angles are in geometric progression.
1986 IMO Longlists, 25
Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations:
\[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\]
Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$
1995 IMC, 2
Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$,
we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
1988 AMC 12/AHSME, 26
Suppose that $p$ and $q$ are positive numbers for which \[ \log_{9}(p) = \log_{12}(q) = \log_{16}(p+q) \] What is the value of $\frac{q}{p}$?
$\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{8}{5}\qquad\textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(E)}\ \frac{16}{9} $
2003 Belarusian National Olympiad, 6
a) A positive integer is called [i]nice [/i] if it can be represented as an arithmetic mean of some (not necessarily distinct) positive integers each being a nonnegative power of $2$.
Prove that all positive integers are nice.
b) A positive integer is called [i]ugly [/i] if it can not be represented as an arithmetic mean of some pairwise distinct positive integers each being a nonnegative power of $2$.
Prove that there exist infinitely many ugly positive integers.
(A. Romanenko, D. Zmeikov)
1976 IMO, 3
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
2024 Vietnam National Olympiad, 1
For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$.
A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$.
b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$
2024 Turkey MO (2nd Round), 1
Let $n\ge3$ be a positive integer. Each edge of a complete graph $K_n$ is assigned a real number satisfying the following conditions:
$(i)$ For any three vertices, the numbers assigned to two of the edges among them are equal, and the number on the third edge is strictly greater.
$(ii) $ The weight of a vertex is defined as the sum of the numbers assigned to the edges emanating from that vertex. The weights of all vertices are equal.
Find all possible values of $n$.
2016 Oral Moscow Geometry Olympiad, 1
The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the altitudes $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the midpoint of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.
1991 Tournament Of Towns, (282) 2
Each of three given circles with radii $1$, $r$ and $r$ touches the others from the outside. For what values of $r$ does there exist a triangle “circumscribed” to these circles? (This means the circles lie inside the triangle, each circle touching two sides of the triangle and each side of the triangle touching two circles.)
(N.B. Vasiliev, Moscow)
2014 District Olympiad, 4
Let $n\geq2$ be a positive integer. Determine all possible values of the sum
\[ S=\left\lfloor x_{2}-x_{1}\right\rfloor +\left\lfloor x_{3}-x_{2}\right\rfloor+...+\left\lfloor x_{n}-x_{n-1}\right\rfloor \]
where $x_i\in \mathbb{R}$ satisfying $\lfloor{x_i}\rfloor=i$ for $i=1,2,\ldots n$.
LMT Team Rounds 2021+, B4
Set $S$ contains exactly $36$ elements in the form of $2^m \cdot 5^n$ for integers $ 0 \le m,n \le 5$. Two distinct elements of $S$ are randomly chosen. Given that the probability that their product is divisible by $10^7$ is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a +b$.
[i]Proposed by Ada Tsui[/i]
2021 Brazil National Olympiad, 1
Let \(ABCD\) be a convex quadrilateral in the plane and let \(O_{A}, O_{B}, O_{C}\) and \(O_{D}\) be the circumcenters of the triangles \(BCD, CDA, DAB\) and \(ABC\), respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.
1997 Baltic Way, 4
Prove that the arithmetic mean $a$ of $x_1,\ldots ,x_n$ satisfies
\[ (x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2\]
PEN H Problems, 79
Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]
1967 Putnam, B5
Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.