We say that a four-digit number $\overline{abcd}$ is [i]slippery [/i] if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?

Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$. \[x^2 + xy = a^2 + ab\] \[y^2 + xy = a^2 - ab\]

Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?

Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.

Two people take turns writing natural numbers from $1$ to $1000$. On the first move, the first player writes the number $1$ on the board. Then with your next move you can write either the number $2a$ or the number $a+1$ on the board if number $a$ is already written on the board. In this case, it is forbidden to write down numbers that are already written on the board. The one who writes out wins the number $1000$ on the board. Who wins if played correctly?

Let \( \mathbb{R} \) be the set of real numbers. Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for any real numbers \( x \) and \( y \), \[ f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1. \]

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

The problems are really difficult to find online, so here are the problems. P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form. a) Determine the smallest possible value of $k$. b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$. c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4. I added this because my translation is a bit weird. [hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana. a) Tentukan bilangan $k$ terkecil yang mungkin. b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$. c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide] P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs. [hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description: For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0). Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve. Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide] Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$. a) Draw the graph of the function $g \circ f$. b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$. P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$. P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$. P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.

Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation.

How many sets of positive integers $(a,b,c)$ satisfy $a>b>c>0$ and $a+b+c=103$?

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.

The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.

The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$ over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$

Show that one can find $50$ distinct positive integers such that the sum of each number and its digits is the same.

A fair $6$-sided die is repeatedly rolled until a $1, 4, 5$, or $6$ is rolled. What is the expected value of the product of all the rolls?

A boy write three times the natural number $n$ in a blackboard. He then performed an operation of the following type several times: He erased one of the numbers and wrote in its place the sum of the two others minus $1$. After several moves, one of the three numbers in the blackboard is $900$. Find all the posible values of $n$.

Let $\{x_n\}^\infty_{n=0}$ be the sequence such that $x_0=2$, $x_1=1$ and $x_{n+2}$ is the remainder of the number $x_{n+1}+x_n$ divided by $7$. Prove that $x_n$ is the remainder of the number $$4^n\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}2\binom n{2k}5^k$$

Two circles $\alpha{}$ and $\beta{}$ with centers $A{}$ and $B{}$ respectively intersect at points $C{}$ and $D{}$. The segment $AB{}$ intersects $\alpha{}$ and $\beta{}$ at points $K{}$ and $L{}$ respectively. The ray $DK$ intersects the circle $\beta{}$ for the second time at the point $N{}$, and the ray $DL$ intersects the circle $\alpha{}$ for the second time at the point $M{}$. Prove that the intersection point of the diagonals of the quadrangle $KLMN$ coincides with the incenter of the triangle $ABC$. [i]Konstantin Knop[/i]

Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.

Let $\omega$ be a circle with diameter $\overline{AB},$ center $O,$ and cyclic quadrilateral $ABCD$ inscribed in it, with $C$ and $D$ on the same side of $\overline{AB}.$ Let $AB=20, BC=13, AD=7.$ Let $\overleftrightarrow{BC}$ and $\overleftrightarrow{AD}$ intersect at $E.$ Let the $E$-excircle of $ECD$ have its center at $L.$ Find $OL.$

The incenter $O$ of an isosceles triangle $ABC$ with $AB=AC$ meets $BC$, $CA$, $AB$ at $K$, $L$, $M$ respectively. Let $N$ be the intersection of lines $OL$ and $KM$ and let $Q$ be the intersection of lines $BN$ and $CA$. Let $P$ be the foot of the perpendicular from $A$ to $BQ$. If we assume that $BP=AP+2PQ$, what are the possible values of $\frac{AB}{BC}$?

A traveller visited a village whose inhabitants either always tell the truth or always lie. The villagers stood in a circle facing the centre of the circle, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction? (B, Frenkin)