Found problems: 85335
2003 District Olympiad, 3
(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that
\[ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . \]
(b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that
\[ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . \]
[i]Dan Ştefan Marinescu, Viorel Cornea[/i]
1992 Austrian-Polish Competition, 3
For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$.
2022 Latvia Baltic Way TST, P5
Let $n \ge 2$ be a positive integer. An $n\times n$ grid of squares has been colored as a chessboard. Let a [i]move[/i] consist of picking a square from the board and then changing the colors to the opposite for all squares that lie in the same row as the chosen square, as well as for all squares that lie in the same column (the chosen square itself is also changed to the opposite color). Find all values of $n$ for which it is possible to make all squares of the grid be the same color in a finite sequence of moves.
2019 PUMaC Geometry B, 1
Suppose we have a convex quadrilateral $ABCD$ such that $\angle B = 100^\circ$ and the circumcircle of $\triangle ABC$ has a center at $D$. Find the measure, in degrees, of $\angle D$.
[i]Note:[/i] The circumcircle of a $\triangle ABC$ is the unique circle containing $A$, $B$, and $C$.
2009 Math Prize For Girls Problems, 13
The figure below shows a right triangle $ \triangle ABC$.
[asy]unitsize(15);
pair A = (0, 4);
pair B = (0, 0);
pair C = (4, 0);
draw(A -- B -- C -- cycle);
pair D = (2, 0);
real p = 7 - 3sqrt(3);
real q = 4sqrt(3) - 6;
pair E = p + (4 - p)*I;
pair F = q*I;
draw(D -- E -- F -- cycle);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);[/asy]
The legs $ \overline{AB}$ and $ \overline{BC}$ each have length $ 4$. An equilateral triangle $ \triangle DEF$ is inscribed in $ \triangle ABC$ as shown. Point $ D$ is the midpoint of $ \overline{BC}$. What is the area of $ \triangle DEF$?
1992 All Soviet Union Mathematical Olympiad, 572
Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.
2013 Putnam, 3
Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that:
(i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and
(ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$
Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$
1997 India National Olympiad, 4
In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$, show that $\dfrac{14}{10} < p < \dfrac{15}{10}$.
2006 Macedonia National Olympiad, 1
A natural number is written on the blackboard. In each step, we erase the units digit and add four times the erased digit to the remaining number, and write the result on the blackboard instead of the initial number. Starting with the number $13^{2006}$, is it possible to obtain the number $2006^{13}$ by repeating this step finitely many times?
2016 Tournament Of Towns, 3
Let $M$ be the midpoint of the base $AC$ of an isosceles $\triangle ABC$. Points $E$ and $F$ on the sides $AB$ and $BC$ respectively are chosen so that $AE \neq CF$ and $\angle FMC = \angle MEF = \alpha$.
Determine $\angle AEM$. [i](6 points) [/i]
[i]Maxim Prasolov[/i]
2013 NIMO Summer Contest, 5
A point $(a,b)$ in the plane is called [i]sparkling[/i] if it also lies on the line $ax+by=1$. Find the maximum possible distance between two sparkling points.
[i]Proposed by Evan Chen[/i]
1979 Austrian-Polish Competition, 6
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$
2008 China Second Round Olympiad, 3
For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying
(1)$0=f(0)<f(1)<f(2)<\ldots$;
(2)$f(n)$ has a finite limit when $n$ approaches infinity;
(3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.
2022 Dutch IMO TST, 4
Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that
One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.
2003 All-Russian Olympiad Regional Round, 11.2
On the diagonal $AC$ of a convex quadrilateral $ABCD$ is chosen such a point $K$ such that $KD = DC$, $\angle BAC = \frac12 \angle KDC$, $\angle DAC = \frac12 \angle KBC$. Prove that $\angle KDA = \angle BCA$ or $\angle KDA = \angle KBA$.
2017 CIIM, Problem 3
Let $G$ be a finite abelian group and $f :\mathbb{Z}^+ \to G$ a completely multiplicative function (i.e. $f(mn) = f(m)f(n)$ for any positive integers $m, n$). Prove that there are infinitely many positive integers $k$ such that $f(k) = f(k + 1).$
2021 ELMO Problems, 5
Let $n$ and $k$ be positive integers. Two infinite sequences $\{s_i\}_{i\geq 1}$ and $\{t_i\}_{i\geq 1}$ are [i]equivalent[/i] if, for all positive integers $i$ and $j$, $s_i = s_j$ if and only if $t_i = t_j$. A sequence $\{r_i\}_{i\geq 1}$ has [i]equi-period[/i] $k$ if $r_1, r_2, \ldots $ and $r_{k+1}, r_{k+2}, \ldots$ are equivalent.
Suppose $M$ infinite sequences with equi-period $k$ whose terms are in the set $\{1, \ldots, n\}$ can be chosen such that no two chosen sequences are equivalent to each other. Determine the largest possible value of $M$ in terms of $n$ and $k$.
2005 National High School Mathematics League, 6
Set $T=\{0,1,2,3,4,5,6\},M=\left\{\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4}|a_i\in T,i=1,2,3,4\right\}$. Put all elements in $M$ in order: from small to large, then the 2005th number is
$\text{(A)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{3}{7^4}$
$\text{(B)}\frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{2}{7^4}$
$\text{(C)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{4}{7^4}$
$\text{(D)}\frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{3}{7^4}$
2007 Purple Comet Problems, 11
A dart board looks like three concentric circles with radii of 4, 6, and 8. Three darts are thrown at the board so that they stick at three random locations on then board. The probability that one dart sticks in each of the three regions of the dart board is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2005 Estonia National Olympiad, 2
Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.
2001 Korea - Final Round, 2
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.
2002 Estonia National Olympiad, 4
Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.
2004 Denmark MO - Mohr Contest, 4
Find all sets $x,y,z$ of real numbers that satisfy
$$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$
2017 Estonia Team Selection Test, 2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2017 Korea Junior Math Olympiad, 2
Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.