Found problems: 85335
In quadrilateral $ABCD$, $AD || BC$, diagonals $AC$ and $BD$ are perpendicular to each other, $X$ and $Y$ are mid-points of $AB$ and $CD$ respectively. Prove that $AB + CD \geq AD + BC$.
Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1,\dots,n\}$ such that $k^4+(\tau(k))^4$ is prime for all $k=1,\dots,n$. Show that $S(n)$ is always a square.
Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$
$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.
Assuming that all of the roots of the cubic equation $x^3 + ax^2 +bx + c = 0$ are real, show that the difference between the greatest and the least roots is not less than $(a^2 - 3b)^{1/2}$ or greater than $2 (a^2 - 3b)^{1/2} / 3^{1/2}.$
$P$ is an arbitrary point inside acute triangle $ABC$. Let $A_1,B_1,C_1$ be the reflections of point $P$ with respect to sides $BC,CA,AB$. Prove that the centroid of triangle $A_1B_1C_1$ lies inside triangle $ABC$.
Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.
$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy.
[i]K. Kokhas[/i]
[b]EDIT.[/b] It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.
Let $ABCD$ be a square. The line segment $AB$ is divided internally at $H$ so that $|AB|\cdot |BH|=|AH|^2$. Let $E$ be the midpoints of $AD$ and $X$ be the midpoint of $AH$. Let $Y$ be a point on $EB$ such that $XY$ is perpendicular to $BE$. Prove that $|XY|=|XH|$.
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and
\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]
for all $x,y\in\mathbb{R}$.
[i]
Proposed by usjl[/i]
$ \triangle ABC$ is a right isosceles triangle. Choose points $ K$ and $ M$ from the hypotenuse $ AB$, such that $ K \in AM$ and $ \angle KCM \equal{} 45$º. Prove that $ (AK)^2 \plus{} (MB)^2 \equal{} (KM)^2$
Thanks for any help.
In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
$\textbf{(A)}\ 35\qquad
\textbf{(B)}\ 40\qquad
\textbf{(C)}\ 48\qquad
\textbf{(D)}\ 56\qquad
\textbf{(E)}\ 72$
Find the greatest natural number $n$ such that $n\leq 2008$ and \[(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]\] is a perfect square.
a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that
the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers.
b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers.
Prove that $a$ and $b$ are integers.
In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).
Each of the integers from 1 to 4027 has been colored either green or red. Changing the color of a number is making it red if it was green and making it green if it was red. Two positive integers $m$ and $n$ are said to be [i]cuates[/i] if either $\frac{m}{n}$ or $\frac{n}{m}$ is a prime number. A [i]step[/i] consists in choosing two numbers that are cuates and changing the color of each of them. Show it is possible to apply a sequence of steps such that every integer from 1 to 2014 is green.
Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs.
$(1)$ Calculate $a(m,n)$ and $b(m,n).$
$(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$
Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients.
a) Prove that it has a vector space structure over the field of numbers rational.
b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space.
c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.
Let $S$ be the set of all positive integers $n$ satisfying the following two conditions:
$\bullet$ $n$ is relatively prime to all positive integers less than or equal to $\frac{n}{6}$
$\bullet$ $2^n \equiv 4$ mod $n$
What is the sum of all numbers in $S$?
If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is:
$\textbf{(A)}\ 16:9 \qquad
\textbf{(B)}\ 9:16 \qquad
\textbf{(C)}\ 4:3 \qquad
\textbf{(D)}\ 3:4 \qquad
\textbf{(E)}\ \text{None of these} $
Prove that the number $19^{1976} + 76^{1976}$:
$(a)$ is divisible by the (Fermat) prime number $F_4 = 2^{2^4} + 1$;
$(b)$ is divisible by at least four distinct primes other than $F_4$.
The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$.
Prove that $P,N$ and $S$ are collinear.
Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?
In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i]
[i]Proposed by Soviet Union.[/i]