Found problems: 85335
Talithia throws a party on the fifth Saturday of every month that has five Saturdays. That is, if a month has five Saturdays, Talithia has a party on the fifth Saturday of that month, and if a month has four Saturdays, then Talithia does not have a party that month. Given that January $1$, $2010$ was a Friday, compute the number of parties Talithia will have in $2010$.
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.
Let $A=(a_{ij})$ be a $5 \times 5$ matrix with $a_{ij}=\min\{i,j\}$. Suppose $f:\mathbb{R}^5 \to \mathbb{R}^5$ is a smooth map such that $f(\Sigma) \subset \Sigma$, where $\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}$. Denote by $f^{(n)}$ te $n$-th iterate of $f$. Prove that there does not exist $N \ge 1$ such that
\[\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.\]
Prove that there exists a set $ S$ of $ n \minus{} 2$ points inside a convex polygon $ P$ with $ n$ sides, such that any triangle determined by $3$ vertices of $ P$ contains exactly one point from $ S$ inside or on the boundaries.
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.
Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these values of $a$.
Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that
\[PA^2 + PB^2 \geq 2r^2.\]
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.
Let $A{}$ and $B$ be invertible $n\times n$ matrices with real entries. Suppose that the inverse of $A+B^{-1}$ is $A^{-1}+B$. Prove that $\det(AB)=1$. Does this property hold for $2\times 2$ matrices with complex entries?
$ABCD$ is convex quadrilateral. If $W_a$ is product of power of $A$ about circle $BCD$ and area of triangle $BCD$. And define $W_b,W_c,W_d$ similarly.prove $W_a+W_b+W_c+W_d=0$
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots + a^{1!} + 1.\]
[i]Proposed by Milos Milosavljevic[/i]
Let $ABC$ be a right triangle with hypotenuse $BC$ and center $I$. Let bisectors of the angles $\angle B$ and $\angle C$ intersect the sides $AC$ and $AB$ in$ D$ and $E$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars of the points $D$ and $E$ on the side $BC$. Prove that $I$ is the circumcenter of $APQ$.
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are such that $AD=AE = BC$. Let $H$ be the common point of the altitudes of triangle $ABC$.
It is known that $AH^{2}=BH^{2}+CH^{2}$.
Prove that $H$ lies on the segment $DE$.
[i]Proposed by D. Shiryaev[/i]
Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$).
Prove that $(S(n))^3 <n^4$.
On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points.
$a)$ Prove that there exist two students with exactly the same answers to problems
$b)$ Prove that there exist at least $4$ students with same number of points
In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.
A $\text{2.25 kg}$ mass undergoes an acceleration as shown below. How much work is done on the mass?
[asy]
// Code by riben
size(350);
// Axes
draw((0,0)--(12,0),lightgray);
draw((0,-3)--(0,5));
// Tick Marks
draw((2,0)--(2,-0.2));
label("2",(2,-0.2),S*2);
draw((4,0)--(4,-0.2));
label("4",(4,-0.2),S*2);
draw((6,0)--(6,-0.2));
label("6",(6,-0.2),S*2);
draw((8,0)--(8,-0.2));
label("8",(8,-0.2),S*2);
draw((10,0)--(10,-0.2));
label("10",(10,-0.2),S*2);
draw((12,0)--(12,-0.2));
label("12",(12,-0.2),S*2);
draw((0,-2)--(-0.2,-2));
label("-2",(-0.2,-2),W);
draw((0,0)--(-0.2,0),lightgray);
label("0",(-0.2,0),W);
draw((0,2)--(-0.2,2),lightgray);
label("2",(-0.2,2),W);
draw((0,4)--(-0.2,4));
label("4",(-0.2,4),W);
// Dashed Lines
draw((0,-2)--(12,-2),dashed);
draw((0,2)--(12,2),dashed+lightgray);
draw((0,4)--(12,4),dashed);
draw((2,5)--(2,0.2),dashed);
draw((4,5)--(4,0.2),dashed);
draw((6,5)--(6,0.2),dashed);
draw((8,5)--(8,0.2),dashed);
draw((10,5)--(10,0.2),dashed);
draw((12,5)--(12,0.2),dashed);
draw((2,-1)--(2,-3),dashed);
draw((4,-1)--(4,-3),dashed);
draw((6,-1)--(6,-3),dashed);
draw((8,-1)--(8,-3),dashed);
draw((10,-1)--(10,-3),dashed);
draw((12,-1)--(12,-3),dashed);
// Path
path A=(0,0)--(2,4)--(4,4)--(6,2)--(8,0)--(10,-2)--(12,0);
draw(A,linewidth(1.5));
// Labels
label(scale(0.85)*rotate(90)*"Acceleration (m/s/s)",(0,1),W*7);
label(scale(0.75)*"Position (m)",(11,0),N);
[/asy]
(A) $\text{36 J}$
(B) $\text{22 J}$
(C) $\text{5 J}$
(D)$\text{-17 J}$
(E) $\text{-36 J}$
The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area.
[asy]
size(150);
defaultpen(linewidth(0.8));
path square=unitsquare;
draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square);
[/asy]
Functions $f$ and $g$ are defined on the set of all integers in the interval $[-100; 100]$ and take integral values. Prove that for some integral $k$ the number of solutions of the equation $f(x)-g(y)=k$ is odd.\\ ( A. Golovanov)