For real numbers $ x_1, x_2, \dots, x_n $ and $ y_1, y_2, \dots, y_n $ , the inequalities hold $ x_1 \geq x_2 \geq \ldots \geq x_n> 0 $ and $$ y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n. $$ Prove that $ ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n $.

Solve the system of equation $$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$

Find the smallest positive integer $n$, such that there exist $n$ integers $x_1, x_2, \dots , x_n$ (not necessarily different), with $1\le x_k\le n$, $1\le k\le n$, and such that \[x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2},\quad\text{ and }x_1x_2 \cdots x_n = n!,\] but $\{x_1, x_2, \dots , x_n\} \ne \{1, 2, \dots , n\}$.

Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent . a) Prove that there are consistent sequences other than $a_n = n$. b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ? c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?

A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer and no points are awarded or deducted for each unanswered question. Joana obtained $87$ points. How many questions did she not answer?

Let \(P(x)=x^2+\dfrac x 2 +b\) and \(Q(x)=x^2+cx+d\) be two polynomials with real coefficients such that \(P(x)Q(x)=Q(P(x))\) for all real \(x\). Find all real roots of \(P(Q(x))=0\).

Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.

Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that $$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$ for all real numbers $x,y$

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$

Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).

For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]

The temperatures $f^o F$ and $c^o C$ are equal when $f = \frac95 c + 32$. What temperature is the same in both $^o F$ and $^o C$?

Let $(G, \cdot )$ be a group and $e$ an identity element. Prove that if all elements $x$ of $G$ satisfy $x\cdot x = e$ then $(G, \cdot)$ is abelian (that is, commutative).

Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it. I know this problem was easy, but it still appeared on our TST, and so I posted it here.

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call a function $\mathbb Z_{>0}\rightarrow \mathbb Z_{>0}$ $\emph{M-rugged}$ if it is unbounded and satisfies the following two conditions: $(1)$ If $f(n)|f(m)$ and $f(n)<f(m)$ then $n|m$. $(2)$ $|f(n+1)-f(n)|\leq M$. a. Find all $1-rugged$ functions. b. Determine if the number of $2-rugged$ functions is smaller than $2019$.

Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.

It is given that $x$ varies directly as $y$ and inversely as the square of $z$, and that $x=10$ when $y=4$ and $z=14$. Then, when $y=16$ and $z=7$, $x$ equals: $ \textbf{(A)}\ 180\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 154\qquad\textbf{(D)}\ 140\qquad\textbf{(E)}\ 120 $

Find $A$ so that the ratio of $3\frac23$ to $22$ is the same as the ratio of $7\frac56$ to $A$

[u]Round 5[/u] [i]Each of the three problems in this round depends on the answer to two of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. [/i] [b]p13.[/b] Let $B$ be the answer to problem $14$, and let $C$ be the answer to problem $15$. A quadratic function $f(x)$ has two real roots that sum to $2^{10} + 4$. After translating the graph of $f(x)$ left by $B$ units and down by $C$ units, the new quadratic function also has two real roots. Find the sum of the two real roots of the new quadratic function. [b]p14.[/b] Let $A$ be the answer to problem $13$, and let $C$ be the answer to problem $15$. In the interior of angle $\angle NOM = 45^o$, there is a point $P$ such that $\angle MOP = A^o$ and $OP = C$. Let $X$ and $Y$ be the reflections of $P$ over $MO$ and $NO$, respectively. Find $(XY)^2$. [b]p15.[/b] Let $A$ be the answer to problem $13$, and let $B$ be the answer to problem $14$. Totoro hides a guava at point $X$ in a flat field and a mango at point $Y$ different from $X$ such that the length $XY$ is $B$. He wants to hide a papaya at point $Z$ such that $Y Z$ has length $A$ and the distance $ZX$ is a nonnegative integer. In how many different locations can he hide the papaya? [u]Round 6[/u] [b]p16.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $AB = 4$, $CD = 8$, $BC = 5$, and $AD = 6$. Given that point $E$ is on segment $CD$ and that $AE$ is parallel to $BC$, find the ratio between the area of trapezoid $ABCD$ and the area of triangle $ABE$. [b]p17.[/b] Find the maximum possible value of the greatest common divisor of $\overline{MOO}$ and $\overline{MOOSE}$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.) [b]p18.[/b] Suppose that $125$ politicians sit around a conference table. Each politician either always tells the truth or always lies. (Statements of a liar are never completely true, but can be partially true.) Each politician now claims that the two people beside them are both liars. Suppose that the greatest possible number of liars is $M$ and that the least possible number of liars is $N$. Determine the ordered pair $(M,N)$. [u]Round 7[/u] [b]p19.[/b] Define a [i]lucky [/i] number as a number that only contains $4$s and $7$s in its decimal representation. Find the sum of all three-digit lucky numbers. [b]p20.[/b] Let line segment $AB$ have length $25$ and let points $C$ and $D$ lie on the same side of line $AB$ such that $AC = 15$, $AD = 24$, $BC = 20$, and $BD = 7$. Given that rays $AC$ and $BD$ intersect at point $E$, compute $EA + EB$. [b]p21.[/b] A $3\times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is $30$, how many distinct grids are possible? [u]Round 8[/u] [b]p22.[/b] Define a sequence of positive integers $s_1, s_2, ... , s_{10}$ to be [i]terrible [/i] if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: $\bullet$ $s_i > s_j $ $\bullet$ $j - i + 1$ divides the quantity $s_i + s_{i+1} + ... + s_j$ Determine the minimum possible value of $s_1 + s_2 + ...+ s_{10}$ over all terrible sequences. [b]p23.[/b] The four points $(x, y)$ that satisfy $x = y^2 - 37$ and $y = x^2 - 37$ form a convex quadrilateral in the coordinate plane. Given that the diagonals of this quadrilateral intersect at point $P$, find the coordinates of $P$ as an ordered pair. [b]p24.[/b] Consider a non-empty set of segments of length $1$ in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3$-[i]amazing [/i] if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a $3$-amazing set of segments. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2934024p26255963]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence of integers $a_1,a_2,a_3,\ldots$ is called [i]exact[/i] if $a_n^2-a_m^2=a_{n-m}a_{n+m}$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.

Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$, \[f(x^2-y^2)=(x-y)(f(x)+f(y))\]

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.