Found problems: 85335
2023 Centroamerican and Caribbean Math Olympiad, 4
A four-digit number $n=\overline{a b c d}$, where $a, b, c$ and $d$ are digits, with $a \neq 0$, is said to be [i]guanaco[/i] if the product $\overline{a b} \times \overline{c d}$ is a positive divisor of $n$. Find all guanaco numbers.
1992 Irish Math Olympiad, 2
If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.
2020 China Northern MO, BP3
Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.
1991 AMC 12/AHSME, 6
If $x \ge 0$, then $\sqrt{x \sqrt{x \sqrt{x}}} = $
$ \textbf{(A)}\ x\sqrt{x}\qquad\textbf{(B)}\ x\sqrt[4]{x}\qquad\textbf{(C)}\ \sqrt[8]{x}\qquad\textbf{(D)}\ \sqrt[8]{x^{3}}\qquad\textbf{(E)}\ \sqrt[8]{x^{7}} $
1994 Putnam, 3
Show that if the points of an isosceles right triangle of side length $1$ are each colored with one of four colors, then there must be two points of the same color which are at least a distance $2-\sqrt 2$ apart.
2005 Bosnia and Herzegovina Team Selection Test, 6
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
2018 Costa Rica - Final Round, A1
If $x \in R-\{-7\}$, determine the smallest value of the expression
$$\frac{2x^2 + 98}{(x + 7)^2}$$
2010 Middle European Mathematical Olympiad, 12
We are given a positive integer $n$ which is not a power of two. Show that ther exists a positive integer $m$ with the following two properties:
(a) $m$ is the product of two consecutive positive integers;
(b) the decimal representation of $m$ consists of two identical blocks with $n$ digits.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 8)[/i]
2016 Purple Comet Problems, 10
Mildred the cow is tied with a rope to the side of a square shed with side length 10 meters. The rope is attached to the shed at a point two meters from one corner of the shed. The rope is 14 meters long. The area of grass growing around the shed that Mildred can reach is given by $n\pi$ square meters, where $n$ is a positive integer. Find $n$.
2023 CCA Math Bonanza, L2.2
For a positive integer $n$ let $f(n)$ denote the number of ways to put $n$ objects into pairs if the only thing that matters is which object each object gets paired with. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023.
[i]Lightning 2.2[/i]
2013 VTRMC, Problem 1
Let $I=3\sqrt2\int^x_0\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$. If $0<x<\pi$ and $\tan I=\frac2{\sqrt3}$, what is $x$?
2018 Purple Comet Problems, 14
A complex number $z$ whose real and imaginary parts are integers satis fies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$
.
2011 Postal Coaching, 2
Let $\tau(n)$ be the number of positive divisors of a natural number $n$, and $\sigma(n)$ be their sum. Find the largest real number $\alpha$ such that
\[\frac{\sigma(n)}{\tau(n)}\ge\alpha \sqrt{n}\]
for all $n \ge 1$.
2001 District Olympiad, 1
a) Find all the integers $m$ and $n$ such that
\[9m^2+3n=n^2+8\]
b) Let $a,b\in \mathbb{N}^*$ . If $x=a^{a+b}+(a+b)^a$ and $y=a^a+(a+b)^{a+b}$ which one is bigger?
[i]Florin Nicoara, Valer Pop[/i]
1986 Tournament Of Towns, (125) 7
Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance.
(A . K . Tolpugo , Kiev)
2003 Czech And Slovak Olympiad III A, 5
Show that, for each integer $z \ge 3$, there exist two two-digit numbers $A$ and $B$ in base $z$, one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$. For instance, in base $10$ these numbers are $A = 18, B = 81$.
Geometry Mathley 2011-12, 5.2
Let $ABCD$ be a rectangle and $U, V$ two points of its circumcircle. Lines $AU,CV$ intersect at $P$ and lines $BU,DV$ intersect at $Q$, distinct from $P$. Prove that $$\frac{1}{PQ^2} \ge \frac{1}{UV^2} - \frac{1}{AC^2}$$
Michel Bataille
2020 Ukrainian Geometry Olympiad - April, 2
Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
2014 Junior Balkan Team Selection Tests - Moldova, 1
Prove that $$\frac{2 }{2013 +1} +\frac{2^{2}}{2013^{2^{1}}+1} +\frac{2^{3}}{2013^{2^{2}}+1} + ...+ \frac{2^{2014}}{2013^{2^{2013}}+1} < \frac{1}{1006}$$
1949 Putnam, B4
Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion
$$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$
are positive integers.
2012 Today's Calculation Of Integral, 826
Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.
2025 Macedonian Balkan MO TST, 1
A set of $n \ge 2$ light bulbs are arranged around a circle, and are consecutively numbered with
$1, 2, . . . , n$. Each bulb can be in one of two states: either it is [b]on[/b] or [b]off[/b]. In the initial configuration,
at least one bulb is turned on. On each one of $n$ days we change the current on/off configuration as
follows: for $1 \le k \le n$, on the $k$-th day we start from the $k$-th bulb and moving in clockwise direction
along the circle, we change the state of every traversed bulb until we switch on a bulb which was
previously off.
Prove that the final configuration, reached on the $n$-th day, coincides with the initial one.
2013 IFYM, Sozopol, 7
Let $n\in \mathbb{N}$. Prove that
$lcm [1,2,..,n]=lcm [\binom{n}{1},\binom{n}{2},...,\binom{n}{n}]$
if and only if $n+1$ is a prime number.
2018 Peru Cono Sur TST, 4
Consider the numbers
$$ S_1 = \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 3} + \frac{1}{1 \cdot 4} + \dots + \frac{1}{1 \cdot 2018}, $$
$$ S_2 = \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 4} + \frac{1}{2 \cdot 5} + \dots + \frac{1}{2 \cdot 2018}, $$
$$ S_3 = \frac{1}{3 \cdot 4} + \frac{1}{3 \cdot 5} + \frac{1}{3 \cdot 6} + \dots + \frac{1}{3 \cdot 2018}, $$
$$ \vdots $$
$$ S_{2017} = \frac{1}{2017 \cdot 2018}. $$
Prove that the number $ S_1 + S_2 + S_3 + \dots + S_{2017} $ is not an integer.
1998 AMC 12/AHSME, 29
A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$ \textbf{(A)}\ 4.0\qquad
\textbf{(B)}\ 4.2\qquad
\textbf{(C)}\ 4.5\qquad
\textbf{(D)}\ 5.0\qquad
\textbf{(E)}\ 5.6$