Found problems: 85335
Let \( \mathbb{Z^{+}} \) denote the set of all positive integers. Find all functions \( f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} \) such that for every pair of positive integers \( a \) and \( b \), there exists a positive integer \( c \) satisfying:
$$
f(a)f(b) - ab = 2^{c-1} - 1.
$$
Proposed by Matin Yousefi
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
In a triangle $ABC$ with $AB = AC$, let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$. The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$. Find the angles of the triangle $BAC$.
Many calculators have a reciprocal key $\boxed{\frac{1}{x}}$ that replaces the current number displayed with its reciprocal. For example, if the display is $\boxed{00004}$ and the $\boxed{\frac{1}{x}}$ key is depressed, then the display becomes $\boxed{000.25}$. If $\boxed{00032}$ is currently displayed, what is the fewest number of times you must depress the $\boxed{\frac{1}{x}}$ key so the display again reads $\boxed{00032}$?
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$
From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$?
[img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that
$f(km)+f(kn)-f(k)f(mn)\ge 1$
for all $k,m,n\in\mathbb{N}$.
Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form
\[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\]
where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.
Determine all real values of the parameter $a$ for which the equation
\[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression.
We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).
On the plane is drawn a triangle $ABC$ and a circle $\omega$ passing through the vertex $C$, the midpoints of the sides $AC$ and $BC$ and the point of intersection of the medians of the triangle $ABC$. The point $K$ lies on the circle $\omega$ such that $\angle AKB=90^o$. Using only with a ruler, draw a tangent to the circle $\omega$ at point $K$.
Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square.
[i]Proposed by Michael Ren
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.
At the ends of a checkered strip measuring $1 \times 101$ squares there are two chips: on the left is the chip of the first player, on the right is the second. Per turn dares to move his piece in the direction of the opposite edge of the strip by 1, 2, 3 or 4 cells. In this case, you are allowed to jump over opponent's chip, but it is forbidden to place your chip on the same square with her. The first one to reach the opposite edge of the strip wins. Who wins if the game is played correctly: the one who goes first, or him rival?
The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent.
[hide=Diagram][asy]import graph; size(11cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -9.26871978147865, xmax = 19.467150423463277, ymin = -6.150626456647122, ymax = 10.10782642246474; /* image dimensions */
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draw((1.0409487561836381,4.30054785243355)--(0.,0.)--(6.,0.)--cycle, aqaqaq);
/* draw figures */
draw((1.0409487561836381,4.30054785243355)--(0.,0.), uququq);
draw((0.,0.)--(6.,0.), uququq);
draw((6.,0.)--(1.0409487561836381,4.30054785243355), uququq);
draw(circle((3.,1.550104087253063), 3.376806580383107));
draw(circle((1.9303371951242874,1.5188413314630436), 1.5188413314630436));
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draw((-1.2916762981259242,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2"));
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draw((1.0559139088339535,1.4932847901569466)--(3.,-1.8267024931300442), linetype("2 2"));
draw((12.047991949367804,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2"));
draw((1.0226422135625703,7.734611112525813)--(-1.2916762981259242,-1.8267024931300444));
draw((-1.2916762981259242,-1.8267024931300444)--(12.047991949367804,-1.8267024931300444));
draw((12.047991949367804,-1.8267024931300444)--(1.0226422135625703,7.734611112525813));
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dot((1.0409487561836381,4.30054785243355),linewidth(3.pt) + dotstyle);
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dot((0.,0.),linewidth(3.pt) + dotstyle);
label("$B$", (-0.5723380089304358,-0.10096957139619551), NE * labelscalefactor);
dot((6.,0.),linewidth(3.pt) + dotstyle);
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dot((1.0226422135625703,7.734611112525813),linewidth(3.pt) + dotstyle);
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dot((-1.2916762981259242,-1.8267024931300444),linewidth(3.pt) + dotstyle);
label("$B_3$", (-1.8146782621516093,-1.4783468086631473), NE * labelscalefactor);
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dot((1.9303371951242874,1.5188413314630436),linewidth(3.pt) + dotstyle);
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dot((1.9303371951242878,0.),linewidth(3.pt) + dotstyle);
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label("$c_2$", (1.5072315453745722,3.247947632939138), NE * labelscalefactor);
dot((2.9254299438737803,2.666303492733126),linewidth(3.pt) + dotstyle);
label("$B_1$", (2.8576013858323694,3.1129106488933584), NE * labelscalefactor);
dot((0.45412477306806903,1.8761589424582812),linewidth(3.pt) + dotstyle);
label("$C_1$", (0,2.3296961414278368), NE * labelscalefactor);
dot((1.0559139088339535,1.4932847901569466),linewidth(3.pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy][/hide]
How many real roots does the polynomial $x^5 + x^4 - x^3 - x^2 - 2x - 2$ have?
$
\textbf{a)}\ 1
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 4
\qquad\textbf{e)}\ \text{None of above}
$
A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$
Consider an acute-angled triangle $ABC$ and its circumcircle. Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.
Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that
$\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$
Find all pairs of polynomials $(p, q)$ with integer coefficients that satisfy the equation $$p(x^2) + q(x^2) = p(x)q(x)$$ such that $p$ is of degree $n$ and has $n$ nonnegative real roots (with multiplicity).
$f: (0,1) \rightarrow [0, \infty)$ is zero except at a countable set of points $a_{1}, a_2, a_3, ... $ . Let $b_n = f(a_n)$. Show that if $\sum b_{n}$ converges, then $f$ is differentiable at at least one point. Show that for any sequence $b_{n}$ of non-negative reals with $\sum b_{n} =\infty$ , we can find a sequence $a_{n}$ such that the function $f$ defined as above is nowhere differentiable.
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color.
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?