Found problems: 85335
1993 Chile National Olympiad, 1
There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.
1984 AMC 12/AHSME, 17
A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A = origin, H = (5,0), B = (13,0), C = (5,6.5);
draw(C--A--B--C--H^^rightanglemark(C,H,B,16));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$H$",H,S);
label("$15$",C/2,NW);
label("$16$",(H+B)/2,S);
[/asy]
$\textbf{(A) }120\qquad
\textbf{(B) }144\qquad
\textbf{(C) }150\qquad
\textbf{(D) }216\qquad
\textbf{(E) }144\sqrt5$
2016 Austria Beginners' Competition, 4
Let $ABCDE$ be a convex pentagon with five equal sides and right angles at $C$ and $D$. Let $P$ denote the intersection point of the diagonals $AC$ and $BD$. Prove that the segments $PA$ and $PD$ have the same length.
(Gottfried Perz)
1963 AMC 12/AHSME, 12
Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is:
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 12 \qquad
\textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 9$
2021 SAFEST Olympiad, 6
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2011 Brazil Team Selection Test, 5
Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions:
1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied.
2) For every pair of real numbers $x$ and $y$,
\[ f(xf(y))+yf(x)=xf(y)+f(xy)\]
is satisfied.
2018 Thailand TSTST, 3
Circles $O_1, O_2$ intersects at $A, B$. The circumcircle of $O_1BO_2$ intersects $O_1, O_2$ and line $AB$ at $R, S, T$ respectively. Prove that $TR = TS$
KoMaL A Problems 2017/2018, A. 717
Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$.
Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.
2014 Thailand Mathematical Olympiad, 7
Let $ABCD$ be a convex quadrilateral with shortest side $AB$ and longest side $CD$, and suppose that $AB < CD$. Show that there is a point $E \ne C, D$ on segment $CD$ with the following property:
For all points $P \ne E$ on side $CD$, if we define $O_1$ and $O_2$ to be the circumcenters of $\vartriangle APD$ and $\vartriangle BPE$ respectively, then the length of $O_1O_2$ does not depend on $P$.
2019 Ramnicean Hope, 1
Show that
$$ \frac{a^4}{(a+b)\left( a^2+b^2 \right)} +\frac{b^4}{(b+c)\left( b^2+c^2 \right)} +\frac{c^4}{(c+a)\left( c^2+a^2 \right)}\ge \frac{a+b+c}{4} , $$
for any positive real numbers $ a,b,c. $
[i]Costică Ambrinoc[/i]
2005 Thailand Mathematical Olympiad, 10
What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$?
2019 Teodor Topan, 4
Let be an odd natural number $ n, $ and $ n $ real numbers $ y_1\le y_2\le\cdots\le y_n $ whose sum is $ 0. $ Prove that
$$ (n+2)y_{\frac{n+1}{2}}^2\le y_1^2+y_2^2+\cdots +y_n^2, $$
and specify where equality is attained.
[i]Nicolae Bourbăcuț[/i]
1964 Miklós Schweitzer, 5
Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?
1992 AMC 8, 22
Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?
[asy]
for (int a=1; a <= 4; ++a)
{
draw((a,0)--(a,2));
}
draw((0,0)--(4,0));
draw((0,1)--(5,1));
draw((1,2)--(5,2));
draw((0,0)--(0,1));
draw((5,1)--(5,2));
[/asy]
$\text{(A)}\ 15 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$
2002 Korea Junior Math Olympiad, 3
For square $ABCD$, $M$ is a midpoint of segment $CD$ and $E$ is a point on $AD$ satisfying $\angle BEM = \angle MED$. $P$ is an intersection of $AM$, $BE$. Find the value of $\frac{PE}{BP}$
2018 Saint Petersburg Mathematical Olympiad, 5
$ABCD$ is inscribed quadrilateral. Line, that perpendicular to $BD$ intersects segments $AB$ and $BC$ and rays $DA,DC$ at $P,Q,R,S$ . $PR=QS$. $M$ is midpoint of $PQ$. Prove that $AM=CM$
2016 Ecuador Juniors, 6
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.
1993 Bulgaria National Olympiad, 3
it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with $7$ vertices) with common base $A_1A_2A_3A_4A_5A_6A_7$ and vertices respectively the points $B$ and $C$. The edges $BA_i , CA_i$ $(i = 1,...,7$), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.
2005 Greece JBMO TST, 4
Find all the positive integers $n , n\ge 3$ such that $n\mid (n-2)!$
2023 Purple Comet Problems, 6
Find the least positive integer such that the product of its digits is $8! = 8 \cdot 7 \cdot6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.
2013 F = Ma, 11
A right-triangular wooden block of mass $M$ is at rest on a table, as shown in figure. Two smaller wooden cubes, both with mass $m$, initially rest on the two sides of the larger block. As all contact surfaces are frictionless, the smaller cubes start sliding down the larger block while the block remains at rest. What is the normal force from the system to the table?
$\textbf{(A) } 2mg\\
\textbf{(B) } 2mg + Mg\\
\textbf{(C) } mg + Mg\\
\textbf{(D) } Mg + mg( \sin \alpha + \sin \beta)\\
\textbf{(E) } Mg + mg( \cos \alpha + \cos \beta)$
2017 AMC 12/AHSME, 9
A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$?
$\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$
2005 SNSB Admission, 2
Let $ \lambda $ be the Lebesgue measure in the plane, let $ u,v\in\mathbb{R}^2, $ let $ A\subset\mathbb{R}^2 $ such that $ \lambda (A)>0 $ and let be the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ f(t)=\int_A \chi_{A+tu}\cdot\chi_{A+tv}\cdot d\lambda $$
[b]a)[/b] Show that $ f $ is continuous.
[b]b)[/b] Prove that any Lebesgue measurable subset of the plane that has nonzero Lebesgue measure contains the vertices of an equilateral triangle.
2002 Iran Team Selection Test, 5
A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.
2005 Baltic Way, 4
Find three different polynomials $P(x)$ with real coefficients such that $P\left(x^2 + 1\right) = P(x)^2 + 1$ for all real $x$.