The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? [asy] import graph; size(12.5cm); real lsf=2; pathpen=linewidth(0.5); pointpen=black; pen fp = fontsize(10); pointfontpen=fp; real xmin=-1.33,xmax=11.05,ymin=-9.01,ymax=-0.44; pen ycycyc=rgb(0.55,0.55,0.55); pair A=(1,-6), B=(1,-2), D=(1,-5.8), E=(1,-5.6), F=(1,-5.4), G=(1,-5.2), H=(1,-5), J=(1,-4.8), K=(1,-4.6), L=(1,-4.4), M=(1,-4.2), N=(1,-4), P=(1,-3.8), Q=(1,-3.6), R=(1,-3.4), S=(1,-3.2), T=(1,-3), U=(1,-2.8), V=(1,-2.6), W=(1,-2.4), Z=(1,-2.2), E_1=(1.4,-2.6), F_1=(1.8,-2.6), O_1=(14,-6), P_1=(14,-5), Q_1=(14,-4), R_1=(14,-3), S_1=(14,-2), C_1=(1.4,-6), D_1=(1.8,-6), G_1=(2.4,-6), H_1=(2.8,-6), I_1=(3.4,-6), J_1=(3.8,-6), K_1=(4.4,-6), L_1=(4.8,-6), M_1=(5.4,-6), N_1=(5.8,-6), T_1=(6.4,-6), U_1=(6.8,-6), V_1=(7.4,-6), W_1=(7.8,-6), Z_1=(8.4,-6), A_2=(8.8,-6), B_2=(9.4,-6), C_2=(9.8,-6), D_2=(10.4,-6), E_2=(10.8,-6), L_2=(2.4,-3.2), M_2=(2.8,-3.2), N_2=(3.4,-4), O_2=(3.8,-4), P_2=(4.4,-3.6), Q_2=(4.8,-3.6), R_2=(5.4,-3.6), S_2=(5.8,-3.6), T_2=(6.4,-3.4), U_2=(6.8,-3.4), V_2=(7.4,-3.8), W_2=(7.8,-3.8), Z_2=(8.4,-2.8), A_3=(8.8,-2.8), B_3=(9.4,-3.2), C_3=(9.8,-3.2), D_3=(10.4,-3.8), E_3=(10.8,-3.8); filldraw(C_1--E_1--F_1--D_1--cycle,ycycyc); filldraw(G_1--L_2--M_2--H_1--cycle,ycycyc); filldraw(I_1--N_2--O_2--J_1--cycle,ycycyc); filldraw(K_1--P_2--Q_2--L_1--cycle,ycycyc); filldraw(M_1--R_2--S_2--N_1--cycle,ycycyc); filldraw(T_1--T_2--U_2--U_1--cycle,ycycyc); filldraw(V_1--V_2--W_2--W_1--cycle,ycycyc); filldraw(Z_1--Z_2--A_3--A_2--cycle,ycycyc); filldraw(B_2--B_3--C_3--C_2--cycle,ycycyc); filldraw(D_2--D_3--E_3--E_2--cycle,ycycyc); D(B--A,linewidth(0.4)); D(H--(8,-5),linewidth(0.4)); D(N--(8,-4),linewidth(0.4)); D(T--(8,-3),linewidth(0.4)); D(B--(8,-2),linewidth(0.4)); D(B--S_1); D(T--R_1); D(N--Q_1); D(H--P_1); D(A--O_1); D(C_1--E_1); D(E_1--F_1); D(F_1--D_1); D(D_1--C_1); D(G_1--L_2); D(L_2--M_2); D(M_2--H_1); D(H_1--G_1); D(I_1--N_2); D(N_2--O_2); D(O_2--J_1); D(J_1--I_1); D(K_1--P_2); D(P_2--Q_2); D(Q_2--L_1); D(L_1--K_1); D(M_1--R_2); D(R_2--S_2); D(S_2--N_1); D(N_1--M_1); D(T_1--T_2); D(T_2--U_2); D(U_2--U_1); D(U_1--T_1); D(V_1--V_2); D(V_2--W_2); D(W_2--W_1); D(W_1--V_1); D(Z_1--Z_2); D(Z_2--A_3); D(A_3--A_2); D(A_2--Z_1); D(B_2--B_3); D(B_3--C_3); D(C_3--C_2); D(C_2--B_2); D(D_2--D_3); D(D_3--E_3); D(E_3--E_2); D(E_2--D_2); label("0",(0.52,-5.77),SE*lsf,fp); label("\$ 5",(0.3,-4.84),SE*lsf,fp); label("\$ 10",(0.2,-3.84),SE*lsf,fp); label("\$ 15",(0.2,-2.85),SE*lsf,fp); label("\$ 20",(0.2,-1.85),SE*lsf,fp); label("$\mathrm{Price}$",(-.65,-3.84),SE*lsf,fp); label("$1$",(1.45,-5.95),SE*lsf,fp); label("$2$",(2.44,-5.95),SE*lsf,fp); label("$3$",(3.44,-5.95),SE*lsf,fp); label("$4$",(4.46,-5.95),SE*lsf,fp); label("$5$",(5.43,-5.95),SE*lsf,fp); label("$6$",(6.42,-5.95),SE*lsf,fp); label("$7$",(7.44,-5.95),SE*lsf,fp); label("$8$",(8.43,-5.95),SE*lsf,fp); label("$9$",(9.44,-5.95),SE*lsf,fp); label("$10$",(10.37,-5.95),SE*lsf,fp); label("Month",(5.67,-6.43),SE*lsf,fp); D(A,linewidth(1pt)); D(B,linewidth(1pt)); D(D,linewidth(1pt)); D(E,linewidth(1pt)); D(F,linewidth(1pt)); D(G,linewidth(1pt)); D(H,linewidth(1pt)); D(J,linewidth(1pt)); D(K,linewidth(1pt)); D(L,linewidth(1pt)); D(M,linewidth(1pt)); D(N,linewidth(1pt)); D(P,linewidth(1pt)); D(Q,linewidth(1pt)); D(R,linewidth(1pt)); D(S,linewidth(1pt)); D(T,linewidth(1pt)); D(U,linewidth(1pt)); D(V,linewidth(1pt)); D(W,linewidth(1pt)); D(Z,linewidth(1pt)); D(E_1,linewidth(1pt)); D(F_1,linewidth(1pt)); D(O_1,linewidth(1pt)); D(P_1,linewidth(1pt)); D(Q_1,linewidth(1pt)); D(R_1,linewidth(1pt)); D(S_1,linewidth(1pt)); D(C_1,linewidth(1pt)); D(D_1,linewidth(1pt)); D(G_1,linewidth(1pt)); D(H_1,linewidth(1pt)); D(I_1,linewidth(1pt)); D(J_1,linewidth(1pt)); D(K_1,linewidth(1pt)); D(L_1,linewidth(1pt)); D(M_1,linewidth(1pt)); D(N_1,linewidth(1pt)); D(T_1,linewidth(1pt)); D(U_1,linewidth(1pt)); D(V_1,linewidth(1pt)); D(W_1,linewidth(1pt)); D(Z_1,linewidth(1pt)); D(A_2,linewidth(1pt)); D(B_2,linewidth(1pt)); D(C_2,linewidth(1pt)); D(D_2,linewidth(1pt)); D(E_2,linewidth(1pt)); D(L_2,linewidth(1pt)); D(M_2,linewidth(1pt)); D(N_2,linewidth(1pt)); D(O_2,linewidth(1pt)); D(P_2,linewidth(1pt)); D(Q_2,linewidth(1pt)); D(R_2,linewidth(1pt)); D(S_2,linewidth(1pt)); D(T_2,linewidth(1pt)); D(U_2,linewidth(1pt)); D(V_2,linewidth(1pt)); D(W_2,linewidth(1pt)); D(Z_2,linewidth(1pt)); D(A_3,linewidth(1pt)); D(B_3,linewidth(1pt)); D(C_3,linewidth(1pt)); D(D_3,linewidth(1pt)); D(E_3,linewidth(1pt)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy] $\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 62 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 89 \qquad \textbf{(E)}\ 100$

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?

Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave? $\textbf{(A) }23\qquad\textbf{(B) }27\qquad\textbf{(C) }29\qquad\textbf{(D) }31\qquad\textbf{(E) }33$

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

A circle with diameter $20$ has points $A, B, C, D, E,$ and $F$ equally spaced along its circumference. A second circle is tangent to the lines $AB$ and $AF$ and internally tangent to the circle. If the second circle has diameter $\sqrt{m}+n$ for integers $m$ and $n$, find $m + n.$ [asy] import geometry; size(180); draw(circle((0,0),5)); pair[] p; string[] l={"A","B","C","D","E","F"}; for (int i=0; i<6; ++i){ p.append(new pair[]{dir(i*60+180)*5}); dot(p[i]); label(l[i],p[i],p[i]/3); } draw(p[0]--p[1]^^p[0]--p[5]); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(-60)*90,p[3]--p[3]+(0,-100))}); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(+60)*90,p[3]--p[3]+(0,+100))}); draw(incircle(p[0],p[6],p[7]));[/asy]

[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.

Let a chain denote a row of positive integers which continue infinitely in both directions, such that for each number $n$, the $n$ numbers directly to the left of $n$ yield $n$ distinct remainders upon division by $n$. (a) If a chain has a maximum integer, what are the possible values of that integer? (b) Does there exist a chain which does not have a maximum integer?

Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable. [I]Proposed by Alexandar Ivanov, Bulgaria.[/i]

Show that any triangle can be subdivided into isosceles triangles.

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$?

Equilateral octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is constructed such that $A_1A_3A_5A_7$ is a square of side length $\sqrt{2}$ and $A_2A_4A_6A_8$ is a square of side length 4/3. For each vertex $A_i$ of the octagon, let $B_i$ be the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i-1}A_{i-2}$, where $A_{i-8} = A_i = A_{i+8}$. Compute $[B_1B_2B_3B_4B_5B_6B_7B_8]^2$. [i]2022 CCA Math Bonanza Team Round #9[/i]

[b]Q15.[/b] Determine the greatest value of the sum $M=xy+yz+zx$, where $x,y,z$ are real numbers satisfying the following condition $x^2+2y^2+5z^2=22.$

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$

Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. [list=1] [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ [/list]

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?