Found problems: 85335
2023 Romania JBMO TST, P1
Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$
2004 Regional Olympiad - Republic of Srpska, 2
Let $ABC$ be a triangle, $T$ its centroid and $S$ its incenter.
Prove that the following conditions are equivalent:
(1) line $TS$ is parallel to one side of triangle $ABC$,
(2) one of the sides of triangle $ABC$ is equal to the half-sum of the other two sides.
2013 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.)
[i]Proposed by Ray Li[/i]
2018 Purple Comet Problems, 13
Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations
$x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$.
Find the maximum possible value of $x + 1000y$.
2012 Korea National Olympiad, 4
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $. Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \]
2014 Saudi Arabia BMO TST, 5
Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]
2024 Princeton University Math Competition, B1
Sunay is in the bottom-left square of a checkerboard which is $5$ squares wide (the left-right direction) and $3$ squares tall (the up-down direction). From any square, he may move one square up, one square down, or one square to the right, provided that he does not fall off the checkerboard and provided that he does not revisit a square. How many paths are there for Sunay from the bottom-left square to the top-right square?
2018 CMIMC Number Theory, 6
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.
2015 IFYM, Sozopol, 8
The quadrilateral $ABCD$ is circumscribed around a circle $k$ with center $I$ and $DA\cap CB=E$, $AB\cap DC=F$. In $\Delta EAF$ and $\Delta ECF$ are inscribed circles $k_1 (I_1,r_1)$ and $k_2 (I_2,r_2)$ respectively. Prove that the middle point $M$ of $AC$ lies on the radical axis of $k_1$ and $k_2$.
2016 Saudi Arabia Pre-TST, 2.3
Let $u$ and $v$ be positive rational numbers with $u \ne v$. Assume that there are infinitely many positive integers $n$ with the property that $u^n - v^n$ are integers. Prove that $u$ and $v$ are integers.
2023 Moldova EGMO TST, 8
Prove that the number $1$ can be written as a sum of $2023$ fractions of the form $\frac{1}{k_i}$, where all nonnegative integers $k_i (1\leq i\leq 2023)$ are distinct.
2001 Moldova National Olympiad, Problem 5
Prove that the sum of the numbers $1,2,\ldots,n$ divides their product if and only if $n+1$ is a composite number.
2010 Saint Petersburg Mathematical Olympiad, 6
For positive numbers is true that $$ab+ac+bc=a+b+c$$
Prove $$a+b+c+1 \geq 4abc$$
2016 Tournament Of Towns, 7
A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for :
(a) $N=3$ ([i]4 points)[/i]
(b) $N=4$ ([i]6 points)[/i]
[i]Alexandr Berdnikov [/i]
MIPT student olimpiad spring 2023, 2
Let $A=a_{ij}$ is simetrical real matrix. Prove that :
$\sum_i e^{a_{ii}} \leq tr (e^A)$
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2013 India IMO Training Camp, 1
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that
\[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]
1999 National Olympiad First Round, 8
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
2003 India Regional Mathematical Olympiad, 5
Suppose $P$ is an interior point of a triangle $ABC$ such that the ratios \[ \frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)} , \frac{d(C,AB)}{d(P,AB)} \] are all equal. Find the common value of these ratios. $d(X,YZ)$ represents the perpendicular distance fro $X$ to the line $YZ$.
Today's calculation of integrals, 882
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
2023 VN Math Olympiad For High School Students, Problem 2
Prove that: $3$ symmedians of a triangle are concurrent at a point; the concurrent point is called the [i]Lemoine[/i] point of the given triangle.
2008 IMAC Arhimede, 5
The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$.
$ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t.
$ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.
2014 ELMO Shortlist, 2
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
2018 BMT Spring, Tie 3
Find $$\sum^{k=672}_{k=0} { 2018\choose {3k+2}} \,\, (mod \, 3)$$
1999 Romania National Olympiad, 1
Let $P(x) = 2x^3-3x^2+2$, and the sets:
$$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$ Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements