Found problems: 85335
2004 Serbia Team Selection Test, 3
Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)
2009 Brazil Team Selection Test, 2
Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$.
[i]Proposed by Charles Leytem, Luxembourg[/i]
2003 May Olympiad, 3
Find all pairs of positive integers $(a,b)$ such that $8b+1$ is a multiple of $a$ and $8a+1$ is a multiple of $b$.
2014 Math Prize For Girls Problems, 3
Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?
2002 Hong kong National Olympiad, 4
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
MBMT Guts Rounds, 2015.11
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after the transferring has occurred?
2020 Purple Comet Problems, 5
The diagram below shows square $ABCD$ which has side length $12$ and has the same center as square $EFGH$ which has side length $6$. Find the area of quadrilateral $ABFE$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/8cfedf396cd2e86092c03fa0dcb1fb3c978965.png[/img]
2004 Iran MO (3rd Round), 4
We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.
MBMT Guts Rounds, 2015.18
The first triangle number is $1$; the second is $1 + 2 = 3$; the third is $1 + 2 + 3 = 6$; and so on. Find the sum of the first $100$ triangle numbers.
2007 Postal Coaching, 1
Let $P$ be a point on the circumcircle of a square $ABCD$. Find all integers $n > 0$ such that the sum $$S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n$$ is constant with respect to the point $P$.
1953 AMC 12/AHSME, 3
The factors of the expression $ x^2\plus{}y^2$ are:
$ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\
\textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$
2009 AMC 10, 14
Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw(rotate(90)*p);
draw(rotate(180)*p);
draw(rotate(270)*p);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 \plus{} \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4$
LMT Accuracy Rounds, 2022 S4
Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.
1997 Chile National Olympiad, 5
Let: $ C_1, C_2, C_3 $ three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.
2010 Contests, 2
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
2022 MIG, 13
Consider the numbers $1$ through $6$ numbered on the coins below. Ella takes a coin from each of the three columns. Bella takes a coin from each of the remaining two columns. Cassandra takes the remaining coin. In how many ways could they have taken out the six coins?
[asy]
size(100);
draw(Circle((0,0),0.45));
label("$1$",(0,0));
draw(Circle((0,1),0.45));
label("$2$",(0,1));
draw(Circle((0,2),0.45));
label("$3$",(0,2));
draw(Circle((1,0),0.45));
label("$5$",(1,0));
draw(Circle((1,1),0.45));
label("$4$",(1,1));
draw(Circle((2,0),0.45));
label("$6$",(2,0));
[/asy]
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }20$
2020 Junior Balkan Team Selection Tests-Serbia, 4#
One hundred tennis players took part in a tournament where they played with each other
exactly one game, with no draws. At the end of the tournament a table (ranking) is formed depending on the number of victories. It is known that one tennis player finished the tournament on
$k$-th place and is the only one with that number of victories, and he has beaten every tennis player who is placed above him in the table and lost to anyone ranked weaker than him on the table. Find the smallest value of $k$.
KoMaL A Problems 2022/2023, A. 854
Prove that
\[\sum_{k=0}^n\frac{2^{2^k}\cdot 2^{k+1}}{2^{2^k}+3^{2^k}}<4\]
holds for all positive integers $n$.
[i]Submitted by Béla Kovács, Szatmárnémeti[/i]
1997 German National Olympiad, 3
In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.
1996 India Regional Mathematical Olympiad, 1
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.
2019 Vietnam TST, P1
In a country there are $n\geq 2$ cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions:
i) Every airway can be licensed to exactly one airline.
ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline.
What is the maximum number of airlines that the government can license to satisfy all of these conditions?
2007 Olympic Revenge, 2
Let $a, b, c \in \mathbb{R}$ with $abc = 1$. Prove that
\[a^{2}+b^{2}+c^{2}+{1\over a^{2}}+{1\over b^{2}}+{1\over c^{2}}+2\left(a+b+c+{1\over a}+{1\over b}+{1\over c}\right) \geq 6+2\left({b\over a}+{c\over b}+{a\over c}+{c\over a}+{c\over b}+{b\over c}\right)\]
2001 Saint Petersburg Mathematical Olympiad, 10.6
For any positive integers $n>m$ prove the following inequality:
$$[m,n]+[m+1,n+1]\geq 2m\sqrt{n}$$
As usual, [x,y] denotes the least common multiply of $x,y$
[I]Proposed by A. Golovanov[/i]
2012 Kazakhstan National Olympiad, 1
For a positive reals $ x_{1},...,x_{n} $ prove inequlity:
$ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$
2001 Hong kong National Olympiad, 4
There are $212$ points inside or on a given unit circle. Prove that there are at least $2001$ pairs of points having distances at most $1$.