Found problems: 85335
1995 IMO Shortlist, 3
Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i \equal{} 1, 2, \ldots, n.$ If $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n,$ prove that
\[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]
2009 Harvard-MIT Mathematics Tournament, 3
How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.)
2010 LMT, 14
Al and Bob are joined by Carl and D’Angelo, and they decide to play a team game of Rock Paper Scissors. A game is called [i]perfect[/i] if some two of the four play the same thing, and the other two also play the same thing, but something different. For example, an example of a perfect game would be Al and Bob playing rock, and Carl and D’Angelo playing scissors, but if all four play paper, we do not have a perfect game. What is the probability of a perfect game?
1993 Tournament Of Towns, (384) 2
The square $ PQRS$ is placed inside the square $ABCD$ in such a way that the segments $AP$, $BQ$, $CR$ and $DS$ intersect neither each other nor the square $PQRS$. Prove that the sum of areas of quadrilaterals $ABQP$ and $CDSR$ is equal to the sum of the areas of quadrilaterals $BCRQ$ and $DAPS$.
(Folklore)
2012 LMT, Individual
[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive).
[b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored?
[b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself).
[b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow?
[b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$?
[b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$?
[b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails?
[b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order?
[b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square.
[b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest.
[b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.)
[b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$.
[b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ .
[b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible.
[b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$.
[b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles?
[b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio?
[b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$?
[b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna?
[b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2011 Harvard-MIT Mathematics Tournament, 9
Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.
2007 IberoAmerican, 5
Let's say a positive integer $ n$ is [i]atresvido[/i] if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.
2012 Stars of Mathematics, 1
Let $\ell$ be a line in the plane, and a point $A \not \in \ell$. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ so that $AQ=PQ$ and $\angle PAQ = 45^{\circ}$.
([i]Dan Schwarz[/i])
2015 CCA Math Bonanza, L4.2
Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be points on $BC$ and $DC$ such that $\angle{EAF}=30^\circ$ and $CE=CF$. Determine the length of $BD$.
[i]2015 CCA Math Bonanza Lightning Round #4.2[/i]
2018 Purple Comet Problems, 7
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac14$ of the way to his destination, and in $45$ minutes he will be exactly $\frac13$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
1995 Turkey Team Selection Test, 1
Given real numbers $b \geq a>0$, find all solutions of the system
\begin{align*}
&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.
\end{align*}
2006 Switzerland Team Selection Test, 2
We place randomly the numbers $1,2, \dots ,2006$ around a circle. A move consists of changing two neighbouring numbers. After a limited numbers of moves all the numbers are diametrically opposite to their starting position. Show that we changed at least once two numbers which had the sum $2007$.
Novosibirsk Oral Geo Oly VIII, 2019.4
Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.
2007 Rioplatense Mathematical Olympiad, Level 3, 5
Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments.
Clarification : the vertices of the original triangle are not considered points of intersection or division.
2023 Chile Junior Math Olympiad, 4
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. The points $P$, $Q$, $R$ are chosen on the sides of the segments $AB$, $BC$, $AC$ respectively in such a way that
$$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RA}=\frac25.$$
Find the area of triangle $PQR$.
[img]https://cdn.artofproblemsolving.com/attachments/8/4/6184d66bd3ae23db29a93eeef241c46ae0ad44.png[/img]
2013 ELMO Shortlist, 7
Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear.
[i]Proposed by Michael Kural[/i]
2013 Harvard-MIT Mathematics Tournament, 4
Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*}
2006 Oral Moscow Geometry Olympiad, 5
Equilateral triangles $ABC_1, BCA_1, CAB_1$ are built on the sides of the triangle $ABC$ to the outside. On the segment $A_1B_1$ to the outer side of the triangle $A_1B_1C_1$, an equilateral triangle $A_1B_1C_2$ is constructed. Prove that $C$ is the midpoint of the segment $C_1C_2$.
(A. Zaslavsky)
2016 Benelux, 3
Find all functions $f :\Bbb{ R}\to \Bbb{Z}$ such that $$\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),$$ for all $x, y \in \Bbb{R}.$
2021 Iran RMM TST, 3
We call a polynomial $P(x)=a_dx^d+...+a_0$ of degree $d$ [i]nice[/i] if
$$\frac{2021(|a_d|+...+|a_0|)}{2022}<max_{0 \le i \le d}|a_i|$$
Initially Shayan has a sequence of $d$ distinct real numbers; $r_1,...,r_d \neq \pm 1$. At each step he choose a positive integer $N>1$ and raises the $d$ numbers he has to the exponent of $N$, then delete the previous $d$ numbers and constructs a monic polynomial of degree $d$ with these number as roots, then examine whether it is nice or not. Prove that after some steps, all the polynomials that shayan produces would be nice polynomials
Proposed by [i]Navid Safaei[/i]
2021 MOAA, 1
Evaluate $2\times 0+2\times 1+ 2+0\times 2 +1$.
[i]Proposed by Nathan Xiong[/i]
2009 China Team Selection Test, 3
Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$
2000 Belarus Team Selection Test, 5.2
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.
2021 AIME Problems, 10
Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://imgur.com/1mfBNNL.png[/img]
2006 Swedish Mathematical Competition, 5
In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. Which of the numbers $f(m,n)$ and $g(m,n)$ is larger?