Consider all functions $f:\mathbb{N}\to\mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

Suppose $m$ and $n$ are positive integers such that $m^2+n^2+m$ is divisible by $mn$. Prove that $m$ is a square number.

We are given $2n$ natural numbers \[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\] Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$.

Antonette gets $ 70\%$ on a 10-problem test, $ 80\%$ on a 20-problem test and $ 90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 77 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 83 \qquad \textbf{(E)}\ 87$

Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.

If $a>1$, $b>1$, and $p=\frac{\log_b(\log_ba)}{\log_ba}$, then $a^n$ equals $\textbf {(A) } 1 \qquad \textbf {(B) } b \qquad \textbf {(C) } \log_ab \qquad \textbf {(D) } \log_ba \qquad \textbf {(E) } a^{\log_ba}$

Let $P$ and $Q$ be points taken inside of triangle $ABC$ such that $\angle APB=\angle AQC$ and $\angle APC=\angle AQB$. Circumcircle of $APQ$ intersects $AB$ and $AC$ second time at $K$ and $L$ respectively. Prove that $B,C,L,K$ are concyclic.

Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$ a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$ b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\textstyle\bigcup_{j=0}^6 L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S}\dfrac1{\sqrt{r(C)}}?\] [asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.78)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.78)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.78)); }[/asy] $\textbf{(A) }\dfrac{286}{35}\qquad\textbf{(B) }\dfrac{583}{70}\qquad\textbf{(C) }\dfrac{715}{73}\qquad\textbf{(D) }\dfrac{143}{14}\qquad\textbf{(E) }\dfrac{1573}{146}$

In a triangle $ABC$, let $D$ and $E$ trisect $BC$, so $BD = DE = EC$. Let $F$ be the point on $AB$ such that $\frac{AF}{F B}= 2$, and $G$ on $AC$ such that $\frac{AG}{GC} =\frac12$ . Let $P$ be the intersection of $DG$ and $EF$, and extend $AP$ to intersect $BC$ at a point $X$. Find $\frac{BX}{XC}$

We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24$? [i]2020 CCA Math Bonanza Lightning Round #1.1[/i]

6. If $ p,q$ are rationals, $r=p+\sqrt{7}q$, then prove there exists a matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})$ for which $\frac{ar+b}{cr+d}=r$ and $det(A)=1$

Let an acute-angled triangle $ABC$'s circumcircle is $S$. $S$'s tangent from $B$ and $C$ intersects at point $M$. A line, lies $M$ and parallel to $[AB]$ intersects with $S$ at points $D$ and $E$, intersect with $[AC]$ at point $F$. Prove that $$[DF]=[FE]$$

Grade8 P5 of Sharygin 2019 Finals

$ABCBE$ is a regular pentagon. Point $B'$ is symmetric to point $B$ wrt line $AC$ (see figure). Is it possible to pave the plane with pentagons equal to $AB'CBE$? (S. Markelov) [img]https://cdn.artofproblemsolving.com/attachments/9/2/cbb5756517e85e56c4a931e761a6b4da8fe547.png[/img]