Found problems: 85335
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
The trapezoid $ABCD$ has sides $AB$ and $CD$ that are parallel $\hat{DAB} = 6^{\circ}$ and $\hat{ABC} = 42^{\circ}$. Point $X$ lies on the side $AB$ , such that $\hat{AXD} = 78^{\circ}$ and $\hat{CXB} = 66^{\circ}$. The distance between $AB$ and $CD$ is $1$ unit . Prove that $AD + DX - (BC + CX) = 8$ units.
(Heir of Ramanujan)
$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$.
[i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b].
[color=#B6D7A8]#1733[/color]
Removing a unit square from a $2\times 2$ square we get a piece called [i]L-tromino.[/i] From the fourth line of a $7 \times 7$ cheesboard some unit squares have been removed. The resulting chessboard is cut in L-trominos. Determine the number and location of the removed squares.
How many ways are there to permute the letters $\{S,C,R, A,M,B,L,E\}$ without the permutation containing the substring $L AME$?
Eminem is trying to find the real Slim Shady in a row of $2025$ indistinguishable Slim Shady clones, one of which is the real Slim Shady. Eminem randomly guesses, and if he guesses wrong, a new clone joins the row and all the clones randomly rearrange themselves. He keeps guessing as more identical clones are added, trying to find the real Slim Shady. Find the probability that he will eventually find him within $15$ guesses.
Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.
For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is
$\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$
We call a graph with n vertices $k-flowing-chromatic$ if:
1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.
2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors.
3. after some action of step 2 we can make all the chess reach each of the n vertices.
Let T(G) denote the least number k such that G is k-flowing-chromatic.
If such k does not exist, denote T(G)=0.
denote $\chi (G)$ the chromatic number of G.
Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.
Given the polynomial $$P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}.$$
Express the numerical value of its derivative of order $325$ for $x = 0$.
Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is
[list=1]
[*] 283
[*] 307
[*] 593
[*] 691
[/list]
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.
The Proenc has a new $8\times 8$ chess board and requires composing it into rectangles that do not overlap, so that:
(i) each rectangle has as many white squares as black ones;
(ii) there are no two rectangles with the same number of squares.
Determines the maximum value of $n$ for which such a decomposition is possible. For this value of $n$, determine all possible sets ${A_1,... ,A_n}$, where $A_i$ is the number of rectangle $i$ in squares, for which a decomposition of the board under the conditions intended actions is possible.
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?
Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq x, y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.
On the plane we mark $n$ ($n > 2$) straight lines passing through one point $O$ in such a way that for any two of them there is a marked straight line that bisects one of the pairs of vertical angles, formed by these straight lines. Prove that the drawn straight lines divide full angle into equal parts.
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is
[asy]
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
draw((1,-1)--(2,-1)--(2,2)--(1,2)--cycle);[/asy]
$ \text{(A)}\ 20\qquad\text{(B)}\ 21\qquad\text{(C)}\ 22\qquad\text{(D)}\ 24\qquad\text{(E)}\ 30 $
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
\[\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4\]
holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices:
1) $T_3$ and $T_3 (A_2)$
2) $T_k$ and $T_k (A_1) $
3) $T_k$ and $T_{k+1}$
Find all integers $a$, $b$, $c$, $d$, and $e$ such that
\begin{align*}a^2&=a+b-2c+2d+e-8,\\b^2&=-a-2b-c+2d+2e-6,\\c^2&=3a+2b+c+2d+2e-31,\\d^2&=2a+b+c+2d+2e-2,\\e^2&=a+2b+3c+2d+e-8.\end{align*}
Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal.
[img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]
A staircase has $100$ steps. Kolya wishes to descend the staircase by alternately jumping down some steps and then up some. The possible jumps he can do are through $6$ (i.e. over $5$ and landing on the $6$th) , $7$ or $8$ steps . He also does not wish to land twice on the same step . Can he descend the staircase in this way?
( S . Fomin, Leningrad)
Let $ABC$ be an isosceles triangle with $AB = AC$. Points $D$, $E$ and $F$ are on sides $BC$, $CA $ and $AB$ respectively, such that $\angle FDE =\angle ABC$ and $FE$ is not parallel to $BC$. Prove that $BC$ is tangent to the circumcircle of the triangle $DEF$, if and only if, $D$ is the midpoint of $BC$.