Found problems: 85335
Swiss NMO - geometry, 2010.9
Let $ k$ and $ k'$ two concentric circles centered at $ O$, with $ k'$ being larger than $ k$. A line through $ O$ intersects $ k$ at $ A$ and $ k'$ at $ B$ such that $ O$ seperates $ A$ and $ B$. Another line through $ O$ intersects $ k$ at $ E$ and $ k'$ at $ F$ such that $ E$ separates $ O$ and $ F$.
Show that the circumcircle of $ \triangle{OAE}$ and the circles with diametres $ AB$ and $ EF$ have a common point.
2014 Taiwan TST Round 1, 5
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1993 Nordic, 4
Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$.
a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd.
b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.
2016 CMIMC, 2
Let $ABCD$ be an isosceles trapezoid with $AD=BC=15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid?
2006 AMC 8, 25
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?
[asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle);
draw(card, linewidth(1));
draw(shift(2.5,0)*card, linewidth(1));
draw(shift(5,0)*card, linewidth(1));
label("$44$", (1,1.5));
label("$59$", shift(2.5,0)*(1,1.5));
label("$38$", shift(5,0)*(1,1.5));[/asy]
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
1991 Baltic Way, 5
For any positive numbers $a, b, c$ prove the inequalities
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]
1997 Tournament Of Towns, (549) 3
In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$.
(Folklore)
2011 China National Olympiad, 1
Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$
1980 IMO, 1
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
2012 Princeton University Math Competition, B5
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
1996 IMO Shortlist, 8
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]
2017 Bosnia And Herzegovina - Regional Olympiad, 1
If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$
2019 Purple Comet Problems, 1
The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
[img]https://cdn.artofproblemsolving.com/attachments/6/3/c23510c821c159d31aff0e6688edebc81e2737.png[/img]
2004 Czech and Slovak Olympiad III A, 1
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
2018 CMIMC Team, 10-1/10-2
Find the smallest positive integer $k$ such that $ \underbrace{11\cdots 11}_{k\text{ 1's}}$ is divisible by $9999$.
Let $T = TNYWR$. Circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. The common external tangent $\ell$ to the two circles closer to $Q$ touches $\omega_1$ and $\omega_2$ at $A$ and $B$ respectively. Line $AQ$ intersects $\omega_2$ at $X$ while $BQ$ intersects $\omega_1$ again at $Y$. Let $M$ and $N$ denote the midpoints of $\overline{AY}$ and $\overline{BX}$, also respectively. If $AQ=\sqrt{T}$, $BQ=7$, and $AB=8$, then find the length of $MN$.
2015 BMT Spring, 10
Let $ABC$ be a triangle with points $E, F$ on $CA$, $AB$, respectively. Circle $C_1$ passes through $E, F$ and is tangent to segment $BC$ at $D$. Suppose that $AE = AF = EF = 3$, $BF = 1$, and $CE = 2$. What is $\frac{ED^2}{F D^2}$ ?
1935 Moscow Mathematical Olympiad, 011
In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2007 India IMO Training Camp, 2
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$
2007 Princeton University Math Competition, 10
if $x$, $y$, and $z$ are real numbers such that $ x^2 + z^2 = 1 $ and $ y^2 + 2y \left( x + z \right) = 6 $, find the maximum value of $ y \left( z - x \right) $.
2013 MTRP Senior, 7
Write 11 numbers on a sheet of paper six zeros and five ones. Perform the following operation 10 times: cross out any two numbers, and if they were equal, write another zero on the board. If they were not equal, write a one. Show that no matter which numbers are chosen at each step, the nal number on the board will be a one.
2012 Pre - Vietnam Mathematical Olympiad, 1
For $a,b,c>0: \; abc=1$ prove that
\[a^3+b^3+c^3+6 \ge (a+b+c)^2\]
2017 JBMO Shortlist, NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.
Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
2009 National Olympiad First Round, 20
Let $ A$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is odd. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
Let $ B$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is even. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
$ A \minus{} B \equal{} ?$
$\textbf{(A)}\ 5000 \qquad\textbf{(B)}\ 4640 \qquad\textbf{(C)}\ 3200 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$
2019 IFYM, Sozopol, 4
For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and
$AB=CB$. Prove that $AD=CD$.