Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

What is the value of $$(\sqrt{2}-1)^4+\frac{1}{(\sqrt{2}-1)^4}?$$ $\textbf{(A) }32-16\sqrt{2}\qquad\textbf{(B) }30\qquad\textbf{(C) }34\qquad\textbf{(D) }15+15\sqrt{2}\qquad\textbf{(E) }16+16\sqrt{2}$

Determine all real numbers $ a$ such that there is a function $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ x\plus{}f(y)\equal{}af(y\plus{}f(x))\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

Compute $$\lim_{x\rightarrow0}\frac{\sqrt[5]{\cos x}-\sqrt[3]{\cos x}}{x^2}.$$

Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and \[a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.\] Prove that \[\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2\]

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

In a triangle $ABC$ equality $2BC=AB+AC$ holds. The angle bisector of $\angle BAC$ inteesects $BC$ at $L$. A circle, that is tangent to $AL$ at $L$ and passes through $B$ intersects $AB$ for the second time at $X$. A circle, that is tangent to $AL$ at $L$ and passes through $C$ intersects $AC$ for the second time at $Y$ Find all possible values of $XY:BC$

Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ($E\ne A, F\ne A$). a) Let $K$ be the intersection of $BE$ and $CF$. Prove that $AK,BC,OG$ are concurrent. b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$) of $(I)$. $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$. Assume that $MN$ intersects $(O)$ at $P$ nad $Q$. Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$, the circumcircle $(GPQ)$ always passes through two fixed points.

In a plane, equilateral triangle $ABC$, square $BCDE$, and regular dodecagon $DEFGHIJKLMNO$ each have side length 1 and do not overlap. Find the area of the circumcircle of $\triangle AFN$.

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

In the rectangular parallelepiped in the figure, the lengths of the segments $EH$, $HG$, and $EG$ are consecutive integers. The height of the parallelepiped is $12$. Find the volume of the parallelepiped. [img]https://cdn.artofproblemsolving.com/attachments/6/4/f74e7fed38c815bff5539613f76b0c4ca9171b.png[/img]

If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$ and determine all $a, b$ yielding equality. [i]Proposed by Gottfried Perz[/i]

The vertices of the cube are assigned $1, 2, 3..., 8$ and then each edge we assign the product of the numbers assigned to its two extreme points. Determine the greatest possible the value of the sum of the numbers assigned to all twelve edges of the cube.

In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation. (1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$. (2) Find the volume of the common part of $V_1$ and $V_2$.

Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively. If $\angle BI_1E = 60^o$, what is the measure of $\angle CI_2F$ in degrees?

In a triangle with sides $a,b,c,t_a,t_b,t_c$ are the corresponding medians and $D$ the diameter of the circumcircle. Prove that \[\frac{a^2+b^2}{t_c}+\frac{b^2+c^2}{t_a}+\frac{c^2+a^2}{t_b}\le 6D\]

Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $ [i]Nedelcu Ion[/i]

We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end. [img]https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png[/img]

A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and $$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$ $\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

Let $\triangle ABC$ be such that $AB = 15, BC = 13, CA = 14.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $P$ is on the same side of $AB$ as $C$ and $AP = BP.$ Let $X$ be the foot of the perpendicular from $P$ to $AC.$ Then the length of $AX$ is $\tfrac{m}{n}$ for some relatively prime positive integers $m$ and $n.$ Find $m + n.$

The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.

Let us consider mathematical crossword which we fill with numbers $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that: 1) All digits occur exactly twice 2) $10$ horizontally divides $4$ vertically 3) $4 \cdot$ ($4$ horizontally - $4$ vertically +$5$) equals $1$ vertically 4) $36$ divides $1$ horizontally and $5$ vertically 5) $9$ vertically divides $5$ vertically In how many ways we can solve this mathematical crossword? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC85LzgyNjUzYmNkNTVmNDE1YTg4OWVkNzAzYzE1M2JkZWE0MThiYWY1LnBuZw==&rn=Y3Jvc3N3b3JkLnBuZw==[/img]

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)