Found problems: 85335
2008 Iran MO (3rd Round), 1
Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: \[ n|1^n\plus{}2^n\plus{}\dots\plus{}k^n\]
2009 AMC 10, 18
Rectangle $ ABCD$ has $ AB\equal{}8$ and $ BC\equal{}6$. Point $ M$ is the midpoint of diagonal $ \overline{AC}$, and E is on $ \overline{AB}$ with $ \overline{ME}\perp\overline{AC}$. What is the area of $ \triangle AME$?
$ \textbf{(A)}\ \frac{65}{8} \qquad
\textbf{(B)}\ \frac{25}{3} \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ \frac{75}{8} \qquad
\textbf{(E)}\ \frac{85}{8}$
2018 AMC 8, 20
In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$
[asy]
size(7cm);
pair A,B,C,DD,EE,FF;
A = (0,0); B = (3,0); C = (0.5,2.5);
EE = (1,0);
DD = intersectionpoint(A--C,EE--EE+(C-B));
FF = intersectionpoint(B--C,EE--EE+(C-A));
draw(A--B--C--A--DD--EE--FF,black+1bp);
label("$A$",A,S); label("$B$",B,S); label("$C$",C,N);
label("$D$",DD,W); label("$E$",EE,S); label("$F$",FF,NE);
label("$1$",(A+EE)/2,S); label("$2$",(EE+B)/2,S);
[/asy]
$\textbf{(A) } \frac{4}{9} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{5}{9} \qquad \textbf{(D) } \frac{3}{5} \qquad \textbf{(E) } \frac{2}{3}$
2021 AMC 12/AHSME Fall, 12
For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n$. For example,
\[f(14) = (1 + 2 + 7 + 14) \div 14 = \frac{12}{7}.\]
What is $f(768) - f(384)?$
$\textbf{(A) }\frac{1}{768}\qquad\textbf{(B) }\frac{1}{192}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad\textbf{(E) }\frac{8}{3}$
2022 Turkey Team Selection Test, 7
What is the minimum value of the expression $$xy+yz+zx+\frac 1x+\frac 2y+\frac 5z$$ where $x, y, z$ are positive real numbers?
2013 Stanford Mathematics Tournament, 1
Robin goes birdwatching one day. he sees three types of birds: penguins, pigeons, and robins. $\frac23$ of the birds he sees are robins. $\frac18$ of the birds he sees are penguins. He sees exactly $5$ pigeons. How many robins does Robin see?
2002 ITAMO, 6
We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white.
a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares?
b) Is it possible to colour some squares in such a way that an odd number of them have exactly $4$ adjacent coloured squares and all the remaining coloured squares have exactly $2$ adjacent coloured squares?
c) Is it possible to colour some squares in such a way that an odd number of them have exactly $2$ adjacent coloured squares and all the remaining coloured squares have exactly $4$ adjacent coloured squares?
2013 AMC 10, 14
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have?
$\textbf{(A) }36\qquad
\textbf{(B) }60\qquad
\textbf{(C) }72\qquad
\textbf{(D) }84\qquad
\textbf{(E) }108\qquad$
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
2014 BmMT, Ind. Round
[b]p1.[/b] Compute $17^2 + 17 \cdot 7 + 7^2$.
[b]p2.[/b] You have $\$1.17$ in the minimum number of quarters, dimes, nickels, and pennies required to make exact change for all amounts up to $\$1.17$. How many coins do you have?
[b]p3.[/b] Suppose that there is a $40\%$ chance it will rain today, and a $20\%$ chance it will rain today and tomorrow. If the chance it will rain tomorrow is independent of whether or not it rained today, what is the probability that it will rain tomorrow? (Express your answer as a percentage.)
[b]p4.[/b] A number is called boxy if the number of its factors is a perfect square. Find the largest boxy number less than $200$.
[b]p5.[/b] Alice, Bob, Carl, and Dave are either lying or telling the truth. If the four of them make the following statements, who has the coin?
[i]Alice: I have the coin.
Bob: Carl has the coin.
Carl: Exactly one of us is telling the truth.
Dave: The person who has the coin is male.[/i]
[b]p6.[/b] Vicky has a bag holding some blue and some red marbles. Originally $\frac23$ of the marbles are red. After Vicky adds $25$ blue marbles, $\frac34$ of the marbles are blue. How many marbles were originally in the bag?
[b]p7.[/b] Given pentagon $ABCDE$ with $BC = CD = DE = 4$, $\angle BCD = 90^o$ and $\angle CDE = 135^o$, what is the length of $BE$?
[b]p8.[/b] A Berkeley student decides to take a train to San Jose, stopping at Stanford along the way. The distance from Berkeley to Stanford is double the distance from Stanford to San Jose. From Berkeley to Stanford, the train's average speed is $15$ meters per second. From Stanford to San Jose, the train's average speed is $20$ meters per second. What is the train's average speed for the entire trip?
[b]p9.[/b] Find the area of the convex quadrilateral with vertices at the points $(-1, 5)$, $(3, 8)$, $(3,-1)$, and $(-1,-2)$.
[b]p10.[/b] In an arithmetic sequence $a_1$, $a_2$, $a_3$, $...$ , twice the sum of the first term and the third term is equal to the fourth term. Find $a_4/a_1$.
[b]p11.[/b] Alice, Bob, Clara, David, Eve, Fred, Greg, Harriet, and Isaac are on a committee. They need to split into three subcommittees of three people each. If no subcommittee can be all male or all female, how many ways are there to do this?
[b]p12.[/b] Usually, spaceships have $6$ wheels. However, there are more advanced spaceships that have $9$ wheels. Aliens invade Earth with normal spaceships, advanced spaceships, and, surprisingly, bicycles (which have $2$ wheels). There are $10$ vehicles and $49$ wheels in total. How many bicycles are there?
[b]p13.[/b] If you roll three regular six-sided dice, what is the probability that the three numbers showing will form an arithmetic sequence? (The order of the dice does matter, but we count both $(1,3, 2)$ and $(1, 2, 3)$ as arithmetic sequences.)
[b]p14.[/b] Given regular hexagon $ABCDEF$ with center $O$ and side length $6$, what is the area of pentagon $ABODE$?
[b]p15.[/b] Sophia, Emma, and Olivia are eating dinner together. The only dishes they know how to make are apple pie, hamburgers, hotdogs, cheese pizza, and ice cream. If Sophia doesn't eat dessert, Emma is vegetarian, and Olivia is allergic to apples, how many di erent options are there for dinner if each person must have at least one dish that they can eat?
[b]p16.[/b] Consider the graph of $f(x) = x^3 + x + 2014$. A line intersects this cubic at three points, two of which have $x$-coordinates $20$ and $14$. Find the $x$-coordinate of the third intersection point.
[b]p17.[/b] A frustum can be formed from a right circular cone by cutting of the tip of the cone with a cut perpendicular to the height. What is the surface area of such a frustum with lower radius $8$, upper radius $4$, and height $3$?
[b]p18.[/b] A quadrilateral $ABCD$ is de ned by the points $A = (2,-1)$, $B = (3, 6)$, $C = (6, 10)$ and $D = (5,-2)$. Let $\ell$ be the line that intersects and is perpendicular to the shorter diagonal at its midpoint. What is the slope of $\ell$?
[b]p19.[/b] Consider the sequence $1$, $1$, $2$, $2$, $3$, $3$, $3$, $5$, $5$, $5$, $5$, $5$, $...$ where the elements are Fibonacci numbers and the Fibonacci number $F_n$ appears $F_n$ times. Find the $2014$th element of this sequence. (The Fibonacci numbers are defined as $F_1 = F_2 = 1$ and for $n > 2$, $F_n = F_{n-1}+F_{n-2}$.)
[b]p20.[/b] Call a positive integer top-heavy if at least half of its digits are in the set $\{7, 8, 9\}$. How many three digit top-heavy numbers exist? (No number can have a leading zero.)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Baltic Way, 13
The point $H$ is the orthocentre of a triangle $ABC$, and the segments $AD,BE,CF$ are its altitudes. The points $I_1,I_2,I_3$ are the incentres of the triangles $EHF,FHD,DHE$ respectively. Prove that the lines $AI_1,BI_2,CI_3$ intersect at a single point.
2016 NIMO Problems, 2
Sitting at a desk, Alice writes a nonnegative integer $N$ on a piece of paper, with $N \le 10^{10}$. Interestingly, Celia, sitting opposite Alice at the desk, is able to properly read the number upside-down and gets the same number $N$, without any leading zeros. (Note that the digits 2, 3, 4, 5, and 7 will not be read properly when turned upside-down.) Find the number of possible values of $N$.
[i]Proposed by Yannick Yao[/i]
2002 AMC 12/AHSME, 5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
[asy]unitsize(.3cm);
defaultpen(linewidth(.8pt));
path c=Circle((0,2),1);
filldraw(Circle((0,0),3),grey,black);
filldraw(Circle((0,0),1),white,black);
filldraw(c,white,black);
filldraw(rotate(60)*c,white,black);
filldraw(rotate(120)*c,white,black);
filldraw(rotate(180)*c,white,black);
filldraw(rotate(240)*c,white,black);
filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$
Champions Tournament Seniors - geometry, 2015.3
Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.
2021 USMCA, 17
Let $X_1X_2X_3X_4$ be a quadrilateral inscribed in circle $\Omega$ such that $\triangle{X_1X_2X_3}$ has side lengths $13,14,15$ in some order. For $1 \le i \le 4$, let $l_i$ denote the tangent to $\Omega$ at $X_i$, and let $Y_i$ denote the intersection of $l_i$ and $l_{i+1}$ (indices taken modulo $4$). Find the least possible area of $Y_1Y_2Y_3Y_4$.
2008 IMO, 4
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$
[i]Author: Hojoo Lee, South Korea[/i]
2010 District Olympiad, 2
Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$.
i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian.
ii) Give an example of a non-abelian group with $ G$'s property from the enounce.
2009 Indonesia TST, 2
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
2018 Polish Junior MO First Round, 4
Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Bisectors of $AD$ and $BC$ intersect line segments $BC$ and $AD$ respectively in points $P$ and $Q$. Show that $\angle APD = \angle BQC$.
1991 AMC 12/AHSME, 29
Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown. If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
[asy]
size(170);
defaultpen(linewidth(0.7)+fontsize(10));
pair B=origin, A=(1.5,3*sqrt(3)/2), C=(3,0), D=(1,0), P=B+1.6*dir(B--A), Q=C+1.2*dir(C--A);
draw(B--P--D--B^^P--Q--D--C--Q);
draw(Q--A--P, linetype("4 4"));
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, E);
label("$A'$", D, S);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
$ \textbf{(A)}\ \frac{8}{5}\qquad\textbf{(B)}\ \frac{7}{20}\sqrt{21}\qquad\textbf{(C)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(D)}\ \frac{13}{8}\qquad\textbf{(E)}\ \sqrt{3} $
2020 Cono Sur Olympiad, 6
A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions:
• each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$;
• the sum of the numbers in each line is $5$;
• the sum of the numbers in each column is $5$;
• the sum of the numbers on each diagonal of four squares is $5$;
• the number written in the upper left box of the board is less than or equal to the other numbers
the board;
• when dividing the board into four $2$ × $2$ squares, in each of them the sum of the four
numbers is $5$.
How many $"brasucas"$ boards are there?
2010 VJIMC, Problem 2
Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.
2018 USAMTS Problems, 2:
Given a set of positive integers $R$, we define the [i]friend set[/i] of $R$ to be all positive integers that are divisible by at least one number in $R$. The friend set of $R$ is denoted by $\mathcal{F}(S_1)=\mathcal{F}(S_2)$. Show that $S_1=S_2$.
2002 AIME Problems, 1
Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible?
2008 ITest, 57
Let $a$ and $b$ be the two possible values of $\tan\theta$ given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If $a+b=m/n$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.