Found problems: 85335
The War of $1812$ started with a declaration of war on Thursday, June $18$, $1812$. The peace treaty to end the war was signed $919$ days later, on December $24$, $1814$. On what day of the week was the treaty signed?
$\textbf{(A)}\ \text{Friday} \qquad
\textbf{(B)}\ \text{Saturday} \qquad
\textbf{(C)}\ \text{Sunday} \qquad
\textbf{(D)}\ \text{Monday} \qquad
\textbf{(E)}\ \text{Tuesday} $
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$
Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that
\[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]
Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$
If the sum of the digits of $a$ is 567, what is the value of $n$?
On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?
Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$.
[i](Ahaan Rungta, 2 points)[/i]
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following two properties.
(i) The Riemann integral $\int_a^b f(t) \mathrm dt$ exists for all real numbers $a < b$.
(ii) For every real number $x$ and every integer $n \ge 1$ we have
\[ f(x) = \frac{n}{2} \int_{x-\frac{1}{n}}^{x+\frac{1}{n}} f(t) \mathrm dt. \]
Real numbers $x,y,z, t$ satisfy $x + y + z +t = 0$ and $x^2+ y^2+ z^2+t^2 = 1$.
Prove that $- 1 \le xy + yz + zt + tx \le 0$.
Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$
Each vertex of a regular heptagon ($7$-gon) is colored either red or blue. Find the number of distinct colorings such that no three consecutive vertices have the same color. Two colorings are considered distinct if one cannot be obtained from the other by a rotation of the heptagon.
[i]Individual #8[/i]
$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.
Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD
and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
In the game of Guess the Card, two players each have a $\frac{1}{2}$ chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered $1,2,\dots,16$ clockwise. They participate in an $4$-round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players $1$ and $9$ face each other in the last round is $\frac{m}{n}$ where $m,n$ are positive integers, find $100m+n$.
[i]Proposed by Evan Chen[/i]
Find the number of positive divisors of $20^{22}$ that are perfect squares or perfect cubes.
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square
Quadratics $g(x) = ax^2 + bx + c$ and $h(x) = dx^2 + ex + f$ are such that the six roots of $g,h$, and $g - h$ are distinct real numbers (in particular, they are not double roots) forming an arithmetic progression in some order. Determine all possible values of $a/d$.
Solve in real number the system $x^3=\frac{z}{y}-\frac{2y}{z}, y^3=\frac{x}{z}-\frac{2z}{x}, z^3=\frac{y}{x}-\frac{2x}{y}$
Prove that for any $\lambda > 3$ there is a number $x$ for which $$\sin x + \sin (\lambda x) \ge 1.8.$$
Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions:
a) not a triple of successive members is chosen,
b) at least one of every triple of successive members is chosen,
c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.
A real function $ f$ satisfies $ 4f(f(x))\minus{}2f(x)\minus{}3x\equal{}0$ for all real numbers $ x$. Prove that $ f(0)\equal{}0$.
Given is an acute triangle $ABC$ with incenter $I$ and the incircle touches $BC, CA, AB$ at $D, E, F$. The circle with center $C$ and radius $CE$ meets $EF$ for the second time at $K$. If $X$ is the $C$-excircle touchpoint with $AB$, show that $CX, KD, IF$ concur.
The inscribed circle of the non-isosceles triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint of the $AC$ lie on one straight line.