Found problems: 1679
2005 Iran MO (3rd Round), 4
Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.
2005 India National Olympiad, 1
Let $M$ be the midpoint of side $BC$ of a triangle $ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L,K$ being nearer to $A$
than $L$. If $AK = KL = LM$, prove that the sides of triangle $ABC$ are in the ratio $5 : 10 : 13$ in some order.
1967 AMC 12/AHSME, 34
Points $D$, $E$, $F$ are taken respectively on sides $AB$, $BC$, and $CA$ of triangle $ABC$ so that $AD:DB=BE:CE=CF:FA=1:n$. The ratio of the area of triangle $DEF$ to that of triangle $ABC$ is:
$\textbf{(A)}\ \frac{n^2-n+1}{(n+1)^2}\qquad
\textbf{(B)}\ \frac{1}{(n+1)^2}\qquad
\textbf{(C)}\ \frac{2n^2}{(n+1)^2}\qquad
\textbf{(D)}\ \frac{n^2}{(n+1)^2}\qquad
\textbf{(E)}\ \frac{n(n-1)}{n+1}$
1998 AMC 12/AHSME, 16
The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is
$ \displaystyle \textbf{(A)}\ \sqrt {\frac {a}{b}} \qquad \textbf{(B)}\ \ \frac {a}{b} \qquad \textbf{(C)}\ \ \frac {a^2}{b^2} \qquad \textbf{(D)}\ \ \frac {a \plus{} b}{2b} \qquad \textbf{(E)}\ \ \frac {a^2 \plus{} 2ab}{b^2 \plus{} 2ab}$
[asy]unitsize(2cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
fill(Arc((1/3,0),2/3,0,180)--reverse(Arc((-2/3,0),1/3,180,360))--reverse(Arc((0,0),1,0,180))--cycle,mediumgray);
draw(unitcircle);
draw(Arc((-2/3,0),1/3,360,180));
draw(Arc((1/3,0),2/3,0,180));
label("$a$",(-2/3,0));
label("$b$",(1/3,0));
draw((-2/3+1/15,0)--(-1/3,0),EndArrow(4));
draw((-2/3-1/15,0)--(-1,0),EndArrow(4));
draw((1/3+1/15,0)--(1,0),EndArrow(4));
draw((1/3-1/15,0)--(-1/3,0),EndArrow(4));[/asy]
2005 Argentina National Olympiad, 5
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
1976 IMO Longlists, 15
Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.
1950 AMC 12/AHSME, 5
If five geometric means are inserted between 8 and 5832, the fifth term in the geometric series:
$\textbf{(A)}\ 648 \qquad
\textbf{(B)}\ 832 \qquad
\textbf{(C)}\ 1168 \qquad
\textbf{(D)}\ 1944 \qquad
\textbf{(E)}\ \text{None of these}$
2024 ELMO Shortlist, N3
Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have
$$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$
[i]Allen Wang[/i]
2009 AIME Problems, 4
In parallelogram $ ABCD$, point $ M$ is on $ \overline{AB}$ so that $ \frac{AM}{AB} \equal{} \frac{17}{1000}$ and point $ N$ is on $ \overline{AD}$ so that $ \frac{AN}{AD} \equal{} \frac{17}{2009}$. Let $ P$ be the point of intersection of $ \overline{AC}$ and $ \overline{MN}$. Find $ \frac{AC}{AP}$.
1963 AMC 12/AHSME, 33
Given the line $y = \dfrac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. A possible equation for $L$ is:
$\textbf{(A)}\ y = \dfrac{3}{4}x + 1 \qquad
\textbf{(B)}\ y = \dfrac{3}{4}x\qquad
\textbf{(C)}\ y = \dfrac{3}{4}x -\dfrac{2}{3} \qquad$
$
\textbf{(D)}\ y = \dfrac{3}{4}x -1 \qquad
\textbf{(E)}\ y = \dfrac{3}{4}x + 2$
2000 239 Open Mathematical Olympiad, 5
Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.
2003 AMC 12-AHSME, 6
The second and fourth terms of a geometric sequence are $ 2$ and $ 6$. Which of the following is a possible first term?
$ \textbf{(A)}\ \minus{}\!\sqrt3 \qquad
\textbf{(B)}\ \minus{}\!\frac{2\sqrt3}{3} \qquad
\textbf{(C)}\ \minus{}\!\frac{\sqrt3}{3} \qquad
\textbf{(D)}\ \sqrt3 \qquad
\textbf{(E)}\ 3$
1971 Czech and Slovak Olympiad III A, 6
Let a tetrahedron $ABCD$ and its inner point $O$ be given. For any edge $e$ of $ABCD$ consider the segment $f(e)$ containing $O$ such that $f(e)\parallel e$ and the endpoints of $f(e)$ lie on the faces of the tetrahedron. Show that \[\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.\]
2015 Thailand TSTST, 1
Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.
1994 Tournament Of Towns, (421) 2
Two circles, one inside the other, are given in the plane. Construct a point $O$, inside the inner circle, such that if a ray from $O$ cuts the circles at $A$ and $B$ respectively, then the ratio $OA/OB$ is constant.
(Folklore)
1971 IMO Longlists, 38
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
1992 Tournament Of Towns, (325) 2
Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$.
(A. Parovan)
1998 Finnish National High School Mathematics Competition, 3
Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$
Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$
2006 Stanford Mathematics Tournament, 10
Evaluate: $\sum^{\infty}_{k=1} \tfrac{k}{a^{k-1}}$ for all $|a|<1$.
1966 IMO Longlists, 63
Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$.
[i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that
$ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$,
where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.
1954 Czech and Slovak Olympiad III A, 4
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
2011 AMC 12/AHSME, 6
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$?
$ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 36 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 60 $
2010 Contests, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
1994 China Team Selection Test, 2
An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.
[b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key.
[b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.
2009 Stars Of Mathematics, 3
Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.