Found problems: 85335
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.
A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set.
Determine the smallest integer $k \ge 0$ having the following property:
for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.
Let $\triangle ABC$ be an acute triangle with angles $\alpha, \beta,$ and $\gamma$. Prove that
$$\frac{\cos(\beta-\gamma)}{cos\alpha}+\frac{\cos(\gamma-\alpha)}{\cos \beta}+\frac{\cos(\alpha-\beta)}{\cos \gamma} \geq \frac{3}{2}$$
Prove that for all positive reals $a,b,c$,
\[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]
In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to
$\text{(A)}$ a triangle
$\text{(B)}$ a square
$\text{(C)}$ rectangle, not square
$\text{(D)}$ rhombus, not square
Let $R=(8,6)$. The lines whose equations are $8y=15x$ and $10y=3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which
\[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\]
is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.
On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $\angle MAK$.
Find all real $a$ such that the equation $3^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7$ has a real root.
[hide=Remark] This was the statement given at the contest, but there was actually a typo and the intended equation was $3^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7$, which is much easier.
In a scalene triangle $ABC$,the bisectors of angle $A,B$ intersect their corresponding sides at $K,L$ respectively.$I,O,H$ denote respectively the incenter,circumcenter and orthocenter of triangle $ABC$. Prove that $A,B,K,L,O$ are concyclic iff $KL$ is the common tangent line of the circumcircles of the three triangles $ALI,BHI$ and $BKI$.
For any positive real numbers $a, b, c$, prove that
\[\sum_{cyclic} \frac{(b + c)(a^4 - b^2 c^2 )}{ab + 2bc + ca} \ge 0\]
Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations:
\begin{align*}
ab + cd &= 8,\\
abcd &= 8 + a + b + c + d.
\end{align*}
Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.
(Two cells of the grid are adjacent if they have a common vertex)
Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur?
In a sequence $P_n$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_n$ has an integral root for each $n$?
Find the sum of all positive integers $n$ such that $1+2+\cdots+n$ divides \[15\left[(n+1)^2+(n+2)^2+\cdots+(2n)^2\right].\]
Suppose $f,g \in \mathbb{R}[x]$ are non constant polynomials. Suppose neither of $f,g$ is the square of a real polynomial but $f(g(x))$ is. Prove that $g(f(x))$ is not the square of a real polynomial.
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$.
Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent
Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $BF = BE$ and such that $ED$ is the angle bisector of $\angle BEC$. Prove that $BD = EF$ if and only if $AF = EC$.
Let $x$ be a variable that can take any positive real value. For certain positive real constants $s$ and $t$, the value of $x^2 + \frac{s}{x}$ is minimized at $x = t$, and the value of $t^2\ln(2 + tx) + \frac{1}{x^2}$ is minimized at $x = s$. Compute the ordered pair $(s, t)$.
Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$
[i]Flavian Georgescu[/i]
Let $ p(x) $ be a quadratic polynomial for which :
\[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \]
Prove that:
\[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]
In triangle $ABC$ ,$M,N,K$ are midpoints of sides $BC,AC,AB$,respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$.$MK,MN$ intersect with semicircles in $X,Y$.The tangents to semicircles at $X,Y$ intersect at point $Z$.Prove that $AZ \perp BC$.(Mehdi E'tesami Fard)
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
[asy]
size(200);
import graph; /* this is a label */
Label f;
f.p=fontsize(0);
xaxis(-0.9,20,Ticks(f, 5.0, 5.0));
yaxis(-0.9,20, Ticks(f, 22.0,5.0)); //
real f(real x)
{
return x;
}
draw(graph(f,-1,22),black+linewidth(1));
label("1", (-1,5), black);
label("2", (-1, 10), black);
label("3", (-1, 15), black);
label("4", (-1, 20), black);
dot((5,5), black+linewidth(5));
dot((10,10), black+linewidth(5));
dot((15, 15), black+linewidth(5));
dot((20,20), black+linewidth(5));
label("MINUTES", (11,-5), S);
label(rotate(90)*"MILES", (-5,11), W);[/asy]
$ \textbf{(A) }5\qquad\textbf{(B) }5.5\qquad\textbf{(C) }6\qquad\textbf{(D) }6.5\qquad\textbf{(E) }7 $