Find three-digit numbers such that any its positive integer power ends with the same three digits and in the same order.

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$. (Netherlands)

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? [list] [*] some rotation around a point of line $\ell$ [*] some translation in the direction parallel to line $\ell$ [*] the reflection across line $\ell$ [*] some reflection across a line perpendicular to line $\ell$ [/list] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

Positive real numbers $a$,$b$,$c$ are given such that $abc=1$.Prove that $$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.$$

A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region? [asy] size(125); defaultpen(linewidth(0.8)); path hexagon=(2*dir(0))--(2*dir(60))--(2*dir(120))--(2*dir(180))--(2*dir(240))--(2*dir(300))--cycle; fill(hexagon,lightgrey); for(int i=0;i<=5;i=i+1) { path arc=2*dir(60*i)--arc(2*dir(60*i),1,120+60*i,240+60*i)--cycle; unfill(arc); draw(arc); } draw(hexagon,linewidth(1.8));[/asy] $ \textbf{(A)}\ 27\sqrt{3}-9\pi\qquad\textbf{(B)}\ 27\sqrt{3}-6\pi\qquad\textbf{(C)}\ 54\sqrt{3}-18\pi\qquad\textbf{(D)}\ 54\sqrt{3}-12\pi\qquad\textbf{(E)}\ 108\sqrt{3}-9\pi $

A circle with center $ O$ is tangent to the coordinate axes and to the hypotenuse of the $ 30^\circ$-$ 60^\circ$-$ 90^\circ$ triangle $ ABC$ as shown, where $ AB \equal{} 1$. To the nearest hundredth, what is the radius of the circle? [asy]defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW); label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]$ \textbf{(A)}\ 2.18\qquad \textbf{(B)}\ 2.24\qquad \textbf{(C)}\ 2.31\qquad \textbf{(D)}\ 2.37\qquad \textbf{(E)}\ 2.41$

Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of two preceding ones. It is known that each of the sequences contains at least one number not present in the other sequence. What is the maximum quantity of numbers common for these two sequences? Boris Frenkin

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

In a triangle $ABC$, right­ angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.

Can it happen that $6$ parallelepipeds, no two of which have common points, are placed in space so that there is a point outside of them from which no vertex of a parallelepiped is visible? (The parallelepipeds are not transparent.) (V Proizvolov)

A rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has $20$ regular triangular faces, $30$ square faces, and $12$ regular pentagonal faces, as shown below. How many rotational symmetries does a rhombicosidodecahedron have? [i]2022 CCA Math Bonanza Lightning Round 4.2[/i]

Given an acute-angled $\vartriangle ABC$ with orthocenter $H$ and altitude $CC_1$. Points $D, E$ and $F$ lie on the segments $AC$, $BC$ and $AB$ respectively, so that $DE \parallel AB$ and $EF \parallel AC$. Denote by $Q$ the symmetric point of $H$ wrt to the midpoint of $DE$. Let $BD \cap CF = P$. If $HP \parallel AB$, prove that the points $C_1, D, Q$ and $E$ lie on a circle.

The integer $n$ has exactly six positive divisors, and they are: $1<a<b<c<d<n$. Let $k=a-1$. If the $k$-th divisor (according to above ordering) of $n$ is equal to $(1+a+b)b$, find the highest possible value of $n$.

Given the equation \[ 4x^3 \plus{} 4x^2y \minus{} 15xy^2 \minus{} 18y^3 \minus{} 12x^2 \plus{} 6xy \plus{} 36y^2 \plus{} 5x \minus{} 10y \equal{} 0,\] find all positive integer solutions.

Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline? $ \textbf{(A) }10\qquad\textbf{(B) }16\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40 $

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that: (a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$ (b) For $x\neq 0,\ f(x)>0.$ Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer. $(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\] $(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\] [i]2010 Kyoto University, Master Course in Mathematics[/i]

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

Let $a$ be a real number such that $\frac{1}{a}=a-[a]$. Show that $a$ is irrational. Clarification: The brackets indicate the integer part of the number they enclose.