Found problems: 85335
Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?
Compute
$$\sum_{n=0}^\infty\frac{n+1}{2^n}.$$
Let $ABCDEF$ be a regular hexagon with $P$ as a point in its interior. Prove that of the three values $\tan \angle APD$, $\tan \angle BPE$ and $\tan \angle CPF$, two of them sum to the third one.
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that
$$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$
for any $x;y\in\mathbb{Q}$
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.
a) Find all values of $P$ lying between 1 and 100.
b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent.
[i]Alan Cheng[/i]
Inside a convex quadrilateral $ABCD$ there is a point $P$ such that the triangles $PAB, PBC, PCD, PDA$ have equal areas. Prove that the area of $ABCD$ is bisected by one of the diagonals.
Prove that there exists an absolute constant $c$ such that any set $H$ of $n$ points of the plane in general position can be coloured with $c\log n$ colours in such a way that any disk of the plane containing at least one point of $H$ intersects some colour class of $H$ in exactly one point.
Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.
Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other.
$\emph{Slovakia}$
Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.
[b]a)[/b] Prove the inequality for $ k\equal{}1$.
[b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.
$F_{n}$ denotes the Fibonacci Sequence where $F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3$ Find$$\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}$$
[list=1]
[*] 2016
[*] 2017
[*] 2018
[*] None of these
[/list]
The diagonals of a convex quadrilateral $ABCD$ intersect at a point $O$. Find all angles of this quadrilateral if $\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}$, and $\measuredangle OAD=30^{\circ}$.
A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$
[i]Dan Schwarz, Romania[/i]
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
Suppose $x,y\in\mathbb{Z}$ satisfy
$$y^4+4y^3+28y+8x^3+6y^2+32x+1=(x^2-y^2)(x^2+y^2+24).$$
Find the sum of all possible values of $|xy|$.
Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively.
A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$
Prove that: $ X,\,Y,\,Z$ are collinear.
Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?
Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:
$ \textbf{(A)}\ \text{is none of the following} \qquad
\textbf{(B)}\ \text{cannot be determined from the given data}\\
\textbf{(C)}\ \text{is 20 or 26} \qquad
\textbf{(D)}\ \text{is 14 or 32} \qquad
\textbf{(E)}\ \text{is 8 or 38}$
Let $p$ be an odd prime. A degree $d$ polynomial $f$ with non-negative integer coefficients less than $p$ is called $p-floppy$ if the coefficients of $f(x)f(-x) - f(x^2)$ are all divisible by $p$ and if exactly $d$ entries in the sequence $(f(0), f(1), f(2), \dots , f(p-1))$ are divisible by $p$. How many non-constant $61$-floppy polynomials are there?
The Quadrumland map is a 6 × 6 square where each square cell is either a kingdom or a disputed territory. There are 27 kingdoms and 9 disputed territories. Each disputed territory is claimed by those and only those kingdoms that are neighbouring with it (adjacent by an edge or a vertex). Is it possible that for each disputed territory the numbers of claims are different?
You can discuss your solutions here
The water city of Platense consists of many platforms and bridges between them. Each bridge connects two platforms and there is not two bridges connecting the same two platforms. The mayor wants to switch some bridges by a series of moves in the following way: if there are three platforms $A,B,C$ and bridges $AB$ and $AC$ ([b]no[/b] bridge $BC$), he can switch bridge $AB$ to a bridge $BC$.
A configuration of bridges is [i]good[/i] if it is possible to go to any platfom from any platform using only bridges. Starting in a good configuration, prove that the mayor can reach any other good configuration, whose the quantity of bridges is the same, using the allowed moves.
For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below.
(a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$.
(b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.
In the diagram below, a group of equilateral triangles are joined together by their sides. A parallelogram in the diagram is defined as a parallelogram whose vertices are all at the intersection of two grid lines and whose sides all travel along the grid lines. Find the number of distinct parallelograms in the diagram below.
[asy]
size(3cm);
pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R;
A=(1, 1.73);
B=(2, 3.46);
C=(3, 5.19);
D=(4, 6.92);
E=(5, 8.65);
F=(6, 10.38);
L=(13, 1.73);
K=(12, 3.46);
J=(11, 5.19);
I=(10, 6.92);
H=(9, 8.65);
G=(8, 10.38);
M=(2,0);
N=(4,0);
O=(6,0);
P=(8,0);
Q=(10,0);
R=(12,0);
draw(A--M);
draw(B--N);
draw(C--O);
draw(D--P);
draw(E--Q);
draw(F--R);
draw(A--L);
draw(B--K);
draw(C--J);
draw(D--I);
draw(E--H);
draw(F--G);
draw(M--G);
draw(N--H);
draw(O--I);
draw(P--J);
draw(Q--K);
draw(R--L);
draw(A--F);
draw(G--L);
draw(M--R);
[/asy]
[i]Proposed by Awesome_guy[/i]