Found problems: 85335
2006 Federal Competition For Advanced Students, Part 2, 1
Let $ N$ be a positive integer. How many non-negative integers $ n \le N$ are there that have an integer multiple, that only uses the digits $ 2$ and $ 6$ in decimal representation?
2009 Stanford Mathematics Tournament, 8
Simplify $\sum_{k=1}^{n}\frac{k^2(k - n)}{n^4}$
2007-2008 SDML (Middle School), 7
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers?
2023 UMD Math Competition Part I, #7
Suppose $S = \{1, 2, 3, x\}$ is a set with four distinct real numbers for which the difference between the largest and smallest values of $S$ is equal to the sum of elements of $S.$ What is the value of $x?$
$$
\mathrm a. ~ {-1}\qquad \mathrm b.~{-3/2}\qquad \mathrm c. ~{-2} \qquad \mathrm d. ~{-2/3} \qquad \mathrm e. ~{-3}
$$
2016 Indonesia TST, 4
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
2017 Romania National Olympiad, 2
Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that
[b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
[b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that
$$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
2006 China Team Selection Test, 2
Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.
2019 CCA Math Bonanza, I8
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$, what are all possible values of $a$?
[i]2019 CCA Math Bonanza Individual Round #8[/i]
2019 Romanian Master of Mathematics Shortlist, original P4
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$
1995 IMO, 2
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that
\[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}.
\]
1989 All Soviet Union Mathematical Olympiad, 491
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
2016 JBMO TST - Turkey, 6
Prove that
\[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \]
for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.
VMEO IV 2015, 10.2
Given a triangle $ABC$ with obtuse $\angle A$ and attitude $AH$ with $H \in BC$. Let $E,F$ on $CA$, $AB$ satisfying $\angle BEH = \angle C$ and $\angle CFH = \angle B$. Let $BE$ cut $CF$ at $D$. Prove that $DE = DF$.
2000 Singapore Senior Math Olympiad, 2
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.
2018 JHMT, 6
$\vartriangle ABC$ is inscribed in a unit circle. The three angle bisectors of $A$,$B$,$C$ are extended to intersect the circle at $A_1$,$B_1$,$C_1$, respectively. Find $$\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}.$$
1901 Eotvos Mathematical Competition, 3
Let $a$ and $b$ be two natural numbers whose greatest common divisor is $d$. Prove that exactly $d$ of the numbers $$a, 2a, 3a, ..., (b-1)a, ba$$ is divisible by $b$.
2005 Korea Junior Math Olympiad, 6
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.
2000 Manhattan Mathematical Olympiad, 2
Farmer Jim has an $8$ gallon bucket full with water. He has three empty buckets: $3$ gallons, $5$ gallons and $8$ gallons. How can he get two volumes of water, $4$ gallons each, using only the four buckets?
2009 Irish Math Olympiad, 3
Find all positive integers $n$ for which $n^8+n+1$ is a prime number.
2005 Irish Math Olympiad, 2
Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
Let $ p(x) \equal{} x^6 \plus{} ax^5 \plus{} bx^4 \plus{} cx^3 \plus{} dx^2 \plus{} ex \plus{} f$ be a polynomial such that $ p(1) \equal{} 1, p(2) \equal{} 2, p(3) \equal{} 3, p(4) \equal{} 4, p(5) \equal{} 5,$ and $ p(6) \equal{} 6.$ What is $ p(7)$?
A. 0
B. 7
C. 14
D. 49
E. 727
2004 Iran MO (3rd Round), 17
Let $ p\equal{}4k\plus{}1$ be a prime. Prove that $ p$ has at least $ \frac{\phi(p\minus{}1)}2$ primitive roots.
2010 Today's Calculation Of Integral, 666
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
1983 Iran MO (2nd round), 6
Suppose that
\[f(x)=\{\begin{array}{cc}n,& \qquad n \in \mathbb N , x= \frac 1n\\ \text{} \\x, & \mbox{otherwise}\end{array}\]
[b]i)[/b] In which points, the function has a limit?
[b]ii)[/b] Prove that there does not exist limit of $f$ in the point $x=0.$
1988 Dutch Mathematical Olympiad, 1
The real numbers $x_1,x_2,..., x_n$ and $a_0,a_1,...,a_{n-1}$ with $x_i \ne 0$ for $i \in\{1,2,.., n\}$ are such that
$$(x-x_1)(x-x_2)...(x-x_n)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$
Express $x_1^{-2}+x_2^{-2}+...+ x_n^{-2}$ in terms of $a_0,a_1,...,a_{n-1}$.