Found problems: 85335
2014 PUMaC Geometry A, 3
Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.
1998 China Team Selection Test, 1
Find $k \in \mathbb{N}$ such that
[b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}
n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.
[b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}
n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.
Find all $n$ which satisfies part [b]b.)[/b]
2023 Vietnam National Olympiad, 6
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously.
a) Find $m$ when $n = 8, a = 4 , b = 1$ .
b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$.
c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.
2011 Thailand Mathematical Olympiad, 10
Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$
\begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*}
for all $m,n \in \mathbb{N}$ ?
2018 Romania National Olympiad, 2
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$
For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$
Determine $\min_{f \in \mathcal{F}}I(f).$
[i]Liviu Vlaicu[/i]
2021 Iranian Geometry Olympiad, 5
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
[i]Proposed by Le Viet An, Vietnam[/i]
2024 Polish MO Finals, 6
Let $ABCD$ be a parallelogram. Let $X \notin AC $ lie inside $ABCD$ so that $\angle AXB = \angle CXD = 90^ {\circ}$ and $\Omega$ denote the circumcircle of $AXC$. Consider a diameter $EF$ of $\Omega$ and assume neither $E, \ X, \ B$ nor $F, \ X, \ D$ are collinear. Let $K \neq X$ be an intersection point of circumcircles of $BXE$ and $DXF$ and $L \neq X$ be such point on $\Omega$ so that $\angle KXL = 90^{\circ}$. Prove that $AB = KL$.
2018 China Team Selection Test, 6
Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$
2009 ITAMO, 3
A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.
2007 Italy TST, 1
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?
2009 Today's Calculation Of Integral, 420
Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane.
(1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis.
(2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.
2014 Contests, 1
Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$
1987 IMO, 3
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.
2015 CCA Math Bonanza, T8
Triangle $ABC$ is equilateral with side length $\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$.
[i]2015 CCA Math Bonanza Team Round #8[/i]
2016 Switzerland - Final Round, 4
There are $2016$ different points in the plane. Show that between these points at least $45$ different distances occur.
1998 AIME Problems, 3
The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region?
2021 LMT Fall, 6
Jared has 3 distinguishable Rolexes. Each day, he selects a subset of his Rolexes and wears them on his arm (the order he wears them does not matter). However, he does not want to wear the same Rolex 2 days in a row. How many ways can he wear his Rolexes during a 6 day period?
2006 Singapore Team Selection Test, 2
Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that
$|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$
Prove that
$\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$
1958 AMC 12/AHSME, 47
$ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR \plus{} PS$ is equal to:
[asy]defaultpen(linewidth(.8pt));
unitsize(3cm);
pair D = origin;
pair C = (2,0);
pair B = (2,1);
pair A = (0,1);
pair P = waypoint(B--A,0.2);
pair S = foot(P,D,B);
pair R = foot(P,A,C);
pair F = foot(A,D,B);
pair Q = foot(P,A,F);
pair T = intersectionpoint(P--Q,A--C);
pair X = intersectionpoint(A--C,B--D);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
draw(P--S);
draw(A--F);
draw(P--R);
draw(P--Q);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$P$",P,N);
label("$S$",S,SE);
label("$T$",T,N);
label("$E$",X,SW+SE);
label("$R$",R,SW);
label("$F$",F,SE);
label("$Q$",Q,SW);[/asy]
$ \textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qquad \textbf{(C)}\ PT \plus{} AT\qquad \textbf{(D)}\ AF\qquad \textbf{(E)}\ EF$
2008 Tournament Of Towns, 6
Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infi nitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.
2019 LIMIT Category B, Problem 4
A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^2=4ax$ is always $90^\circ$. The locus of $P$ is
$\textbf{(A)}~\text{a parabola}$
$\textbf{(B)}~\text{a circle}$
$\textbf{(C)}~\text{an ellipse}$
$\textbf{(D)}~\text{a straight line}$
2014 Turkey EGMO TST, 1
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $AD$ intersect the circumcircle of $ABC$ for the second time at $E$. Let $P$ be the point symmetric to the point $E$ with respect to the point $D$ and $Q$ be the point of intersection of the lines $CP$ and $AB$. Prove that if $A,C,D,Q$ are concyclic, then the lines $BP$ and $AC$ are perpendicular.
2009 ITAMO, 1
Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take.
2001 India IMO Training Camp, 1
Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$.
Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$.
[b](a)[/b] Prove that $r_{1}+r_{2}=2r$.
[b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.
2014 Contests, 1
We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )
[i]Proposed by Erfan Salavati[/i]