Found problems: 85335
2013 Junior Balkan Team Selection Tests - Romania, 3
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$
where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$
2014 Singapore Senior Math Olympiad, 19
In a triangle $\triangle ABC$ it is given that $(\sin A+\sin B):(\sin B+\sin C):(\sin C+\sin A)=9:10:11$.
Find the value of $480\cos A$
2021 Durer Math Competition Finals, 14
How many functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 16\}$ have the property that $f(f(x))-4x$ is divisible by $17$ for all integers $1 \le x \le 16$?
1995 Moldova Team Selection Test, 1
Prove that for any positive integers $m{}$ and $n{}$ the number $\sum_{k=1}^{n} cos^{2m} \frac{k\pi}{2n+1}$ is not an integer.
2023 Brazil Cono Sur TST, 4
Let $n$ be a positive integer. Prove that $n\sqrt{19}\{n\sqrt{19}\} > 1$, where $\{x\}$ denotes the fractional part of $x$.
2019 Balkan MO Shortlist, G1
Let $ABCD$ be a square of center $O$ and let $M$ be the symmetric of the point $B$ with respect to point $A$. Let $E$ be the intersection of $CM$ and $BD$, and let $S$ be the intersection of $MO$ and $AE$. Show that $SO$ is the angle bisector of $\angle ESB$.
2011 Kosovo Team Selection Test, 4
From the number $7^{1996}$ we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.
2011 Singapore Junior Math Olympiad, 2
Two circles $\Gamma_1, \Gamma_2$ with radii $r_i, r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $ \Gamma_1$ at $R$ and intersects $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.
Brazil L2 Finals (OBM) - geometry, 2014.4
Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$. If the $(EFGH) = 1$. Find:
$(ABCD) \times (XYZW)$
Note $(P)$ denote the area of the polygon $P$.
1955 AMC 12/AHSME, 3
If each number in a set of ten numbers is increased by $ 20$, the arithmetic mean (average) of the original ten numbers:
$ \textbf{(A)}\ \text{remains the same} \qquad
\textbf{(B)}\ \text{is increased by 20} \qquad
\textbf{(C)}\ \text{is increased by 200} \\
\textbf{(D)}\ \text{is increased by 10} \qquad
\textbf{(E)}\ \text{is increased by 2}$
2009 India National Olympiad, 2
Define a a sequence $ {<{a_n}>}^{\infty}_{n\equal{}1}$ as follows
$ a_n\equal{}0$, if number of positive divisors of $ n$ is [i]odd[/i]
$ a_n\equal{}1$, if number of positive divisors of $ n$ is [i]even[/i]
(The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x\equal{}0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.
1990 Tournament Of Towns, (258) 2
We call a collection of weights (each weighing an integer value) basic if their total weight equals $500$ and each object of integer weight not greater than $500$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equal value are chosen to balance against a particular object, if in fact there is a choice.)
(a) Find an example of a basic collection other than the collection of $500$ weights each of value $1$.
(b) How many different basic collections are there?
(D. Fomin, Leningrad)
2019 Brazil National Olympiad, 1
Let $\omega_1$ and $\omega_2$ be two circles with centers $C_1$ and $C_2$, respectively, which intersect at two points $P$ and $Q$. Suppose that the circumcircle of triangle $PC_1C_2$ intersects $\omega_1$ at $A \neq P$ and $\omega_2$ at $B \neq P$. Suppose further that $Q$ is inside the triangle $PAB$. Show that $Q$ is the incenter of triangle $PAB$.
2015 India IMO Training Camp, 2
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
1950 AMC 12/AHSME, 9
The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is:
$\textbf{(A)}\ r^2 \qquad
\textbf{(B)}\ r^3 \qquad
\textbf{(C)}\ 2r^2 \qquad
\textbf{(D)}\ 2r^3 \qquad
\textbf{(E)}\ \dfrac{1}{2}r^2$
1999 Turkey Team Selection Test, 2
Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.
2003 Tournament Of Towns, 4
In the sequence $00, 01, 02, 03,\ldots , 99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19, 39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?
1986 IMO Longlists, 64
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
2014 Vietnam National Olympiad, 2
Find the maximum of
\[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\]
where $x,y,z$ are positive real numbers.
2023 Belarusian National Olympiad, 8.5
In every cell of the table $3 \times 3$ a monomial with a positive coefficient is written (cells (1,1); (2,3); (3,2) have the degree of two, cells (1,2);(2,1);(3,3) have a degree of one, cells (3,1);(2,2);(1,3) have a constant).
Vuga added up monomials in every row and got three quadratic polynomials. It turned out that exactly $N$ of them have real roots. Leka added up monomials in every column and got three quadratic polynomials. It turned out that exactly $M$ of them have real roots.
Find the maximum possible value of $N-M$.
TNO 2023 Senior, 1
Let \( n \geq 4 \) be an integer. Show that at a party of \( n \) people, it is possible for each person to have greeted exactly three other people if and only if \( n \) is even.
2013 Peru MO (ONEM), 2
The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.
Clarification: gcd stands for greatest common divisor.
2011 Purple Comet Problems, 23
Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$. Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.
1998 German National Olympiad, 5
A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$.
Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.
2013 Iran MO (2nd Round), 2
Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero.
([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)