Found problems: 85335
2010 Contests, 4
(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$.
(b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers.
Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.
2009 Italy TST, 3
Find all pairs of integers $(x,y)$ such that
\[ y^3=8x^6+2x^3y-y^2.\]
1993 Flanders Math Olympiad, 3
For $a,b,c>0$ we have:
\[ -1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1 \]
2020 JBMO TST of France, 2
a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$
b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$
2016 Math Prize for Girls Problems, 2
Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?
1970 Czech and Slovak Olympiad III A, 3
Let $p>0$ be a given parameter. Determine all real $x$ such that \[\frac{1}{\,x+\sqrt{p-x^2\,}\,}+\frac{1}{\,x-\sqrt{p-x^2\,}\,}\ge\frac{1}{\,p\,}.\]
2023 International Zhautykov Olympiad, 1
Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.
2008 AMC 10, 6
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
2014 Dutch IMO TST, 4
Let $\triangle ABC$ be a triangle with $|AC|=2|AB|$ and let $O$ be its circumcenter. Let $D$ be the intersection of the bisector of $\angle A$ with $BC$. Let $E$ be the orthogonal projection of $O$ to $AD$ and let $F\ne D$ be the point on $AD$ satisfying $|CD|=|CF|$. Prove that $\angle EBF=\angle ECF$.
2012 Junior Balkan Team Selection Tests - Romania, 4
$100$ weights, measuring $1,2, ..., 100$ grams, respectively, are placed in the two pans of a scale such that the scale is balanced. Prove that two weights can be removed from each pan such that the equilibrium is not broken.
1989 AMC 12/AHSME, 15
Hi guys,
I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this:
1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though.
2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary.
3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions:
A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh?
B. Do NOT go back to the previous problem(s). This causes too much confusion.
C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for.
4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving!
Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D
2013 Grand Duchy of Lithuania, 2
Let $ABC$ be an isosceles triangle with $AB = AC$. The points $D, E$ and $F$ are taken on the sides $BC, CA$ and $AB$, respectively, so that $\angle F DE = \angle ABC$ and $FE$ is not parallel to $BC$. Prove that the line $BC$ is tangent to the circumcircle of $\vartriangle DEF$ if and only if $D$ is the midpoint of the side $BC$.
2015 AMC 8, 14
Which of the following integers cannot be written as the sum of four consecutive odd integers?
$\textbf{(A)}\text{ 16}\qquad\textbf{(B)}\text{ 40}\qquad\textbf{(C)}\text{ 72}\qquad\textbf{(D)}\text{ 100}\qquad\textbf{(E)}\text{ 200}$
2017 Princeton University Math Competition, A2/B4
Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. (Note that $012$ and $007$ are not $3$-digit numbers.)
2004 Germany Team Selection Test, 1
Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$.
Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero.
[i]Proposed by Kiran Kedlaya, USA[/i]
2016 Azerbaijan Team Selection Test, 3
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.
1996 All-Russian Olympiad Regional Round, 9.1
Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.
2020 Estonia Team Selection Test, 3
With expressions containing the symbol $*$, the following transformations can be performed:
1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$;
2) rewrite the expression in the form $x * 1$ as $x$.
Conversions can only be performed with an integer expression, but not with its parts.
For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$).
Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$
it is possible to lead to a form in which there is not a single asterisk.
Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.
2004 Postal Coaching, 5
How many paths from $(0,0)$ to $(n,n)$ of length $2n$ are there with exactly $k$ steps. A step is an occurence of the pair $EN$ in the path
2007 Danube Mathematical Competition, 3
For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n\minus{}f(n)\equal{}a$ has infinitely many solutions for any positive integer $ a$.
Ukrainian TYM Qualifying - geometry, 2010.16
Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.
2011 Purple Comet Problems, 2
The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy]
2023 Purple Comet Problems, 17
Let $x, y$, and $z$ be positive integers satisfying the following system of equations:
$$x^2 +\frac{2023}{x}= 2y^2$$
$$y +\frac{2028}{y^2} = z^2$$
$$2z +\frac{2025}{z^2} = xy$$
Find $x + y + z$.
2018 IFYM, Sozopol, 2
$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$.
Prove the following inequality:
$xy + yz + zx \geq 3$.
2007 Rioplatense Mathematical Olympiad, Level 3, 2
Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$, respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$, respectively.
If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.