Found problems: 85335
2023 Oral Moscow Geometry Olympiad, 3
In an acute triangle $ABC$ the line $OI$ is parallel to side $BC$. Prove that the center of the nine-point circle of triangle $ABC$ lies on the line $MI$, where $M$ is the midpoint of $BC$.
2002 Iran Team Selection Test, 12
We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic.
[i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.
2007 Estonia Math Open Senior Contests, 2
Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.
1957 Moscow Mathematical Olympiad, 366
Solve the system: $$\begin{cases} \dfrac{2x_1^2}{1+x_1^2}=x_2 \\ \\ \dfrac{2x_2^2}{1+x_2^2}=x_3\\ \\ \dfrac{2x_3^2}{1+x_3^2}=x_1\end{cases}$$
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
MathLinks Contest 3rd, 2
Let n be a positive integer and let $a_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n}$ be $4n - 1$ positive real numbers such that $c^2_{i+j} \ge a_ib_j $, for all $1 \le i, j \le n$. Also let $m = \max_{2 \le i\le 2n} c_i$.
Prove that $$\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{
b_1 + b_2 + ...+ b_n}{n}\right)$$
2021 IMO Shortlist, G4
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
2005 Today's Calculation Of Integral, 1
Calculate the following indefinite integral.
[1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$
[2] $\int \sin x\cos 3x dx$
[3] $\int \sin 2x\sin 3x dx$
[4] $\int \frac{dx}{4x^2-12x+9}$
[5] $\int \cos ^4 x dx$
2016 Bulgaria JBMO TST, 2
The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.
2007 ITest, 24
Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\
\textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23 \end{array}$
2014 International Zhautykov Olympiad, 3
Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
[i]Proposed by Alexander S. Golovanov, Russia[/i]
1988 Tournament Of Towns, (167) 4
The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.
2023 Czech-Polish-Slovak Junior Match, 2
For a positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Determine all positive integers $n$ for which $d(n)$ is the second largest divisor of $n$.
2017 South East Mathematical Olympiad, 7
Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.
2018 Germany Team Selection Test, 2
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2024 Tuymaada Olympiad, 5
Given a board with size $25\times 25$. Some $1\times 1$ squares are marked, so that for each $13\times 13$ and $4\times 4$ sub-boards, there are atleast $\frac{1}{2}$ marked parts of the sub-board. Find the least possible amount of marked squares in the entire board.
2015 District Olympiad, 3
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets?
1994 Romania TST for IMO, 1:
Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$.
We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$).
A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).
(a)Show that the number of even sets equals the number of odd sets.
(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.
(c)Compute the sum of the measures of the odd sets.
1983 IMO Longlists, 33
Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$
2014 IMO Shortlist, N2
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
2010 IMO Shortlist, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2020 SIME, 11
Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.
1953 Miklós Schweitzer, 4
[b]4.[/b] Show that every closed curve c of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. [b](G. 8)[/b]
2010 Canadian Mathematical Olympiad Qualification Repechage, 6
There are $15$ magazines on a table, and they cover the surface of the table entirely. Prove that one can always take away $7$ magazines in such a way that the remaining ones cover at least $\dfrac{8}{15}$ of the area of the table surface
2012 AMC 10, 3
The point in the $xy$-plane with coordinates $(1000,2012)$ is reflected across line $y=2000$. What are the coordinates of the reflected point?
$ \textbf{(A)}\ (998,2012) \qquad\textbf{(B)}\ (1000,1988)\qquad\textbf{(C)}\ (1000,2024)\qquad\textbf{(D)}\ (1000,4012)\qquad\textbf{(E)}\ (1012,2012) $