Found problems: 18
2020 Latvia Baltic Way TST, 11
Circle centred at point $O$ intersects sides $AC, AB$ of triangle $\triangle ABC$ at points $B_1$ and $C_1$ respectively and passes through points $B,C$. It is known that lines $AO, CC_1, BB_1 $ are concurrent. Prove that $\triangle ABC$ is isosceles.
1971 AMC 12/AHSME, 1
The number of digits in the number $N=2^{12}\times 5^8$ is
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad \textbf{(E) }20$
2008 Harvard-MIT Mathematics Tournament, 6
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
2021 AMC 10 Spring, 12
Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
$\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$
2009 AMC 10, 5
What is the sum of the digits of the square of $ 111,111,111$?
$ \textbf{(A)}\ 18 \qquad
\textbf{(B)}\ 27 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 63 \qquad
\textbf{(E)}\ 81$
2011 AMC 12/AHSME, 8
In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A + H$?
$ \textbf{(A)}\ 17 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 25 \qquad
\textbf{(D)}\ 26 \qquad
\textbf{(E)}\ 43
$
2012 AMC 10, 12
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
2023 AMC 12/AHSME, 13
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
2008 Bundeswettbewerb Mathematik, 2
Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.
2020 Latvia Baltic Way TST, 1
Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds:
$$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$
2007 ITest, 13
What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros?
$\textbf{(A) }3\hspace{14em}\textbf{(B) }4\hspace{14em}\textbf{(C) }5$
$\textbf{(D) }6\hspace{14em}\textbf{(E) }7\hspace{14em}\textbf{(F) }8$
$\textbf{(G) }9\hspace{14em}\textbf{(H) }10\hspace{13.3em}\textbf{(I) }11$
$\textbf{(J) }12\hspace{13.8em}\textbf{(K) }13\hspace{13.3em}\textbf{(L) }14$
$\textbf{(M) }2007$
2013 AMC 12/AHSME, 11
Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?
[asy]
size(180);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
real s=1/2,m=5/6,l=1;
pair A=origin,B=(l,0),C=rotate(60)*l,D=(s,0),E=rotate(60)*s,F=m,G=rotate(60)*m;
draw(A--B--C--cycle^^D--E^^F--G);
dot(A^^B^^C^^D^^E^^F^^G);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$D$",D,S);
label("$E$",E,NW);
label("$F$",F,S);
label("$G$",G,NW);
[/asy]
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{3}{2}\qquad
\textbf{(C) }\dfrac{21}{13}\qquad
\textbf{(D) }\dfrac{13}{8}\qquad
\textbf{(E) }\dfrac{5}{3}\qquad$
2012 AMC 12/AHSME, 9
A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born?
$ \textbf{(A)}\ \text{Friday}
\qquad\textbf{(B)}\ \text{Saturday}
\qquad\textbf{(C)}\ \text{Sunday}
\qquad\textbf{(D)}\ \text{Monday}
\qquad\textbf{(E)}\ \text{Tuesday}
$
2023 AMC 10, 16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
2008 Harvard-MIT Mathematics Tournament, 7
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
2013 AMC 8, 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 47 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 49$
2022 AMC 12/AHSME, 11
Let $ f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n $, where $i = \sqrt{-1}$. What is $f(2022)$
$ \textbf{(A)}\ -2 \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \sqrt{3} \qquad
\textbf{(E)}\ 2$
2021 AMC 12/AHSME Spring, 7
Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?
$\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$