Found problems: 85335
2010 Poland - Second Round, 3
The $n$-element set of real numbers is given, where $n \geq 6$. Prove that there exist at least $n-1$ two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.
1988 China Team Selection Test, 1
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:
\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]
Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
2021 China Team Selection Test, 6
Let $n(\ge 2)$ be an integer. $2n^2$ contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that
(1)If A wins B and B wins C, then A wins C.
(2)there are at most $\frac{n^3}{16}$ draws.
Proof that it is possible to choose $n^2$ contestants and label them $P_{ij}(1\le i,j\le n)$, so that for any $i,j,i',j'\in \{1,2,...,n\}$, if $i<i'$, then $P_{ij}$ wins $P_{i'j'}$.
2007 Princeton University Math Competition, 6
Find the number of ordered triplets of nonnegative integers $(m, n, p)$ such that $m+3n+5p \le 600$.
1962 AMC 12/AHSME, 35
A man on his way to dinner short after $ 6: 00$ p.m. observes that the hands of his watch form an angle of $ 110^{\circ}.$ Returning before $ 7: 00$ p.m. he notices that again the hands of his watch form an angle of $ 110^{\circ}.$ The number of minutes that he has been away is:
$ \textbf{(A)}\ 36 \frac23 \qquad
\textbf{(B)}\ 40 \qquad
\textbf{(C)}\ 42 \qquad
\textbf{(D)}\ 42.4 \qquad
\textbf{(E)}\ 45$
2018-2019 SDML (High School), 11
For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is
$ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$
2011 Today's Calculation Of Integral, 722
Find the continuous function $f(x)$ such that :
\[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]
2005 Bosnia and Herzegovina Junior BMO TST, 2
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”
2014-2015 SDML (Middle School), 1
Given that each unit square in the grid below is a $1\times1$ square, find the area of the shaded region in square units.
[asy]
fill((3,0)--(4,0)--(6,3)--(4,4)--(4,3)--(0,2)--(2,2)--cycle, grey);
draw((0,0)--(6,0));
draw((0,1)--(6,1));
draw((0,2)--(6,2));
draw((0,3)--(6,3));
draw((0,4)--(6,4));
draw((0,0)--(0,4));
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
draw((4,0)--(4,4));
draw((5,0)--(5,4));
draw((6,0)--(6,4));
[/asy]
$\text{(A) }8\qquad\text{(B) }9\qquad\text{(C) }10\qquad\text{(D) }11\qquad\text{(E) }12$
Ukraine Correspondence MO - geometry, 2003.11
Let $ABCDEF$ be a convex hexagon, $P, Q, R$ be the intersection points of $AB$ and $EF$, $EF$ and $CD$, $CD$ and $AB$. $S, T,UV$ are the intersection points of $BC$ and $DE$, $DE$ and $FA$, $FA$ and $BC$, respectively. Prove that if $$\frac{AB}{PR}=\frac{CD}{RQ}=\frac{EF}{QP},$$ then $$\frac{BC}{US}=\frac{DE}{ST}=\frac{FA}{TU}.$$
2004 Federal Math Competition of S&M, 1
Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.
2019 China Team Selection Test, 6
Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time.
Determine all $k$ such that $A$ can always win the game.
2005 Bundeswettbewerb Mathematik, 2
Let be $x$ a rational number.
Prove: There are only finitely many triples $(a,b,c)$ of integers with $a<0$ and $b^2-4ac=5$ such that $ax^2+bx+c$ is positive.
2008 National Olympiad First Round, 20
Each of the integers $a_1,a_2,a_3,\dots,a_{2008}$ is at least $1$ and at most $5$. If $a_n < a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an increasing pair. If $a_n > a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an decreasing pair. If the sequence contains $103$ increasing pairs, at least how many decreasing pairs are there?
$
\textbf{(A)}\ 21
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 102
\qquad\textbf{(E)}\ \text{None of the above}
$
2010 Today's Calculation Of Integral, 574
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,
2022 Junior Balkan Team Selection Tests - Romania, P3
Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]
2012 Sharygin Geometry Olympiad, 4
Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.
(B.Frenkin)
2023 Austrian MO National Competition, 4
The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.
TNO 2008 Junior, 8
A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements:
**Person 1:** The red car was guilty, and either the green or the blue one was involved.
**Person 2:** Either the green car or the red car was guilty, but not both.
**Person 3:** Only one of the cars was guilty, but it was not the blue one.
The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied.
Which car(s) were responsible for the accident?
2016 Cono Sur Olympiad, 1
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2006 Hanoi Open Mathematics Competitions, 6
On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?
2022 Czech and Slovak Olympiad III A, 6
Consider any graph with $50$ vertices and $225$ edges. We say that a triplet of its (mutually distinct) vertices is [i]connected[/i] if the three vertices determine at least two edges. Determine the smallest and the largest possible number of connected triples.
[i](Jan Mazak, Josef Tkadlec)[/i]
2023 India Regional Mathematical Olympiad, 6
Consider a set of $16$ points arranged in $4 \times 4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.
1994 French Mathematical Olympiad, Problem 1
For each positive integer $n$, let $I_n$ denote the number of integers $p$ for which $50^n<7^p<50^{n+1}$.
(a) Prove that, for each $n$, $I_n$ is either $2$ or $3$.
(b) Prove that $I_n=3$ for infinitely many $n\in\mathbb N$, and find at least one such $n$.
2011 Purple Comet Problems, 7
Find the prime number $p$ such that $71p + 1$ is a perfect square.