Found problems: 85335
Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]
Find solution in positive integers : $$n^5+n^4=7^m-1$$
Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?
Let $S=\{1, 2, 3, \dots 2021\}$ and $f:S \to S$ be a function such that $f^{(n)}(n)=n$ for each $n \in S$.
Find all possible values for $f(2021)$.
(Here, $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$.)
[i]Authored by Viktor Simjanoski[/i]
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.
Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.
A crate of toys remains at rest on a sleigh as the sleigh is pulled up a hill with an increasing speed. The crate is not fastened down to the sleigh. What force is responsible for the crate’s increase in speed up the hill?
$\textbf{(A)} \ \text{the force of static friction of the sleigh on the crate}$
$ \textbf{(B)} \ \text{the contact force (normal force) of the ground on the sleigh}$
$ \textbf{(C)} \ \text{the contact force (normal force) of the sleigh on the crate}$
$ \textbf{(D)} \ \text{the gravitational force acting on the sleigh}$
$ \textbf{(E)} \ \text{no force is needed}$
Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$
Two integers are called [i]relatively prime[/i] if they share no common factors other than $1.$ Determine the sum of all positive integers less than $162$ that are relatively prime to $162.$
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]
For all $x,y\in\mathbb R$.
Let us denote by $b(n)$ the number of ways to represent $n$ in the form
$$n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k,$$
where the coefficients at, $r = 1$,$2$,$...$, $k$ can be equal to $0$, $1$ or $2$. Find $b(1996)$.
In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
[i]Proposed by Jorge Tipe, Peru[/i]
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?
Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
$ \textbf{(A)}\ 80 \qquad
\textbf{(B)}\ 82 \qquad
\textbf{(C)}\ 85 \qquad
\textbf{(D)}\ 90 \qquad
\textbf{(E)}\ 95 $
Find the minimum value of
\[\frac{9x^2 \sin^2 x + 4}{x \sin x}\]
for $0 < x < \pi$.
In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that
\[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]
Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions:
[b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$
[b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint.
[b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length.
Prove that $ R$ has at least one side of integral length.
[i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
The schematic wiring diagram below is made of six electric resistances $a_1,a_2,a_3,a_4,a_5,a_6(a_1>a_2>a_3>a_4>a_5>a_6)$. How can we choose the electric resistances, so that the all-in resistance takes its minimum value?
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy81LzBiZGVjZjczN2Y4MWY0YmNhMTg1YmQxNGEzZWMwZDc2OTE1NzUwLnBuZw==&rn=MjExMTExMTExMTExMTFnaGoucG5n[/img][/center]
$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings?
(A. Andjans)
In the television program “Field of Miracles,” the presenter played the prize as follows. The player was shown three boxes, one of which contained a prize. The player pointed to one of the boxes, after which the leader opened one of the other two remaining boxes, which turned out to be empty. After this, the player could either insist on the original choice, or change it and choose the third box. In what case does his chance of winning increase? (There are three possible answers: both boxes are equal, it is better to keep the original choice, it is better to change it. Try to justify your answer.)
In the expansion of
\[ \left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{27}\right)\left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{14}\right)^2,
\]what is the coefficient of $ x^{28}$?
$ \textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405$