Found problems: 85335
2015 Peru Cono Sur TST, P4
In a small city there are $n$ bus routes, with $n > 1$, and each route has exactly $4$ stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of $n$.
1968 AMC 12/AHSME, 6
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E$. Let $S$ represent the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC$. If $r=S/S'$, then:
$\textbf{(A)}\ r=1\text{ sometimes, }r>1\text{ sometimes} \qquad\\
\textbf{(B)}\ r=1\text{ sometimes, }r<1\text{ sometimes} \qquad\\
\textbf{(C)}\ 0<r<1\qquad
\textbf{(D)}\ r>1 \qquad
\textbf{(E)}\ r=1 $
2010 Albania National Olympiad, 1
Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle).
Albanian National Mathematical Olympiad 2010---12 GRADE Question 1.
1995 All-Russian Olympiad Regional Round, 10.4
There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.
2016 Taiwan TST Round 1, 1
Let $AB$ be a chord on a circle $O$, $M$ be the midpoint of the smaller arc $AB$. From a point $C$ outside the circle $O$ draws two tangents to the circle $O$ at the points $S$ and $T$. Suppose $MS$ intersects with $AB$ at the point $E$, $MT$ intersects with $AB$ at the point $F$. From $E,F$ draw a line perpendicular to $AB$ that intersects with $OS,OT$ at the points $X,Y$, respectively. Draw another line from $C$ which intersects with the circle $O$ at the points $P$ and $Q$. Let $R$ be the intersection point of $MP$ and $AB$. Finally, let $Z$ be the circumcenter of triangle $PQR$.
Prove that $X$,$Y$ and $Z$ are collinear.
2007 Germany Team Selection Test, 1
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2000 Czech And Slovak Olympiad IIIA, 6
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$
2014 JHMMC 7 Contest, 18
A $6\text{-year stock}$ that goes up $30\%$ in the first year, down $30\%$ in the second, up $30\%$ in the third, down $30\%$ in the fourth, up $30\%$ in the fifth, and down $30\%$ in the sixth is equivalent to a $3\text{-year stock}$ that loses $x\%$ in each of its three years. Compute $x$.
2005 Cuba MO, 3
Determine all the quadruples of real numbers that satisfy the following:
[i]The product of any three of these numbers plus the fourth is constant.[/i]
VMEO II 2005, 4
a) Let $ABC$ be a triangle and a point $I$ lies inside the triangle. Assume $\angle IBA > \angle ICA$ and $\angle IBC >\angle ICB$. Prove that, if extensions of $BI$, $CI$ intersect $AC$, $AB$ at $B'$, $C'$ respectively, then $BB' < CC'$.
b) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Prove that for every point $I, J$ on the segment $[AD]$ and $I \ne J$, we always have $\angle JBI > \angle JCI$.
c) Let $ABC$ be a triangle with $AB < AC$ and angle bisector $AD$. Choose $M, N$ on segments $CD$ and $BD$, respectively, such that $AD$ is the bisector of angle $\angle MAN$. On the segment $[AD]$ take an arbitrary point $I$ (other than $D$). The lines $BI$, $CI$ intersect $AM$, $AN$ at $B', C'$. Prove that $BB' < CC'$.
2017 CCA Math Bonanza, L2.4
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$. Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$.
[i]2017 CCA Math Bonanza Lightning Round #2.4[/i]
2007 IMO Shortlist, 3
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.
[i]Author: Gerhard Wöginger, Netherlands[/i]
2010 Today's Calculation Of Integral, 540
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.
1978 Bundeswettbewerb Mathematik, 3
For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$
2015 Polish MO Finals, 3
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.
2008 IberoAmerican Olympiad For University Students, 6
[i][b]a)[/b][/i] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^2+B^2=C^2$.
[b][i]b)[/i][/b] Determine if there are matrices $A,B,C\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $A^4+B^4=C^4$.
[b]Note[/b]: The notation $A\in \mathrm{SL}_{2}(\mathbb{Z})$ means that $A$ is a $2\times 2$ matrix with integer entries and $\det A=1$.
1999 National Olympiad First Round, 26
Let $ x$, $ y$, $ z$ be integers such that
\[ \begin{array}{l} {x \minus{} 3y \plus{} 2z \equal{} 1} \\
{2x \plus{} y \minus{} 5z \equal{} 7} \end{array}
\]
Then $ z$ can be
$\textbf{(A)}\ 3^{111} \qquad\textbf{(B)}\ 4^{111} \qquad\textbf{(C)}\ 5^{111} \qquad\textbf{(D)}\ 6^{111} \qquad\textbf{(E)}\ \text{None}$
2008 Purple Comet Problems, 1
Find the greatest prime factor of the sum of the two largest two-digit prime numbers.
2010 Chile National Olympiad, 3
The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.
1955 AMC 12/AHSME, 9
A circle is inscribed in a triangle with sides $ 8$, $ 15$, and $ 17$. The radius of the circle is:
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 7$
2005 MOP Homework, 3
Prove that the equation $a^3-b^3=2004$ does not have any solutions in positive integers.
2010 Today's Calculation Of Integral, 526
For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?
2014 USAMTS Problems, 1:
Fill in each blank unshaded cell with a positive integer less than 100, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
size(9cm);
for (int x=0; x<=11; ++x)
draw((x, 0) -- (x, 5), linewidth(.5pt));
for (int y=0; y<=5; ++y)
draw((0, y) -- (11, y), linewidth(.5pt));
filldraw((0,4)--(0,3)--(2,3)--(2,4)--cycle, gray, gray);
filldraw((1,1)--(1,2)--(3,2)--(3,1)--cycle, gray, gray);
filldraw((4,1)--(4,4)--(5,4)--(5,1)--cycle, gray, gray);
filldraw((7,0)--(7,3)--(6,3)--(6,0)--cycle, gray, gray);
filldraw((7,4)--(7,5)--(6,5)--(6,4)--cycle, gray, gray);
filldraw((8,1)--(8,2)--(10,2)--(10,1)--cycle, gray, gray);
filldraw((9,4)--(9,3)--(11,3)--(11,4)--cycle, gray, gray);
draw((0,0)--(11,0)--(11,5)--(0,5)--cycle);
void foo(int x, int y, string n)
{
label(n, (x+0.5, y+0.5));
}
foo(1, 2, "10");
foo(4, 0, "31");
foo(5, 0, "26");
foo(10, 0, "59");
foo(0, 4, "3");
foo(7, 4, "59");
[/asy]
PEN A Problems, 14
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
2011 Sharygin Geometry Olympiad, 19
Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?