Which is more $\log_3 7$ or $\log_{\frac{1}{3}} \frac{1}{7}$ ?

A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$. Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$

Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$. Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.

Find all reals $(a, b, c)$ such that $$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$

Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?

Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.

The polynomial $P (x)$ is such that the polynomials $P (P (x))$ and $P (P (P (x)))$ are strictly monotone on the whole real axis. Prove that $P (x)$ is also strictly monotone on the whole real axis.

Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$

If $x$, $y$, $z$ are real numbers satisfying \begin{align*} (x + 1)(y + 1)(z + 1) & = 3 \\ (x + 2)(y + 2)(z + 2) & = -2 \\ (x + 3)(y + 3)(z + 3) & = -1, \end{align*} find the value of $$ (x + 20)(y + 20)(z + 20). $$

A rocket is launched and reaches $120$ m in height; in the fall he loses $60$ m, then it recovers $40$ m, loses $ 30 $ again, gains $24$, loses $20$, etc. If the process continues indefinitely, at what height does it tend to stabilize?

Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$; (a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$; (b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.

For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$ Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$ Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.

Let $a,b,c$ be integers, $a>0$ and the equation $ax^2-bx+c=0$ has two distinct real roots in the interval $(0,1)$. Find the least possible value of $a$.