Found problems: 85335
1970 AMC 12/AHSME, 31
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, what is the probability that this number is divisible by $11$?
$\textbf{(A) }2/5\qquad\textbf{(B) }1/5\qquad\textbf{(C) }1/6\qquad\textbf{(D) }1/11\qquad \textbf{(E) }1/15$
1960 AMC 12/AHSME, 13
The polygon(s) formed by $y=3x+2$, $y=-3x+2$, and $y=-2$, is (are):
$ \textbf{(A) }\text{An equilateral triangle}\qquad\textbf{(B) }\text{an isosceles triangle} \qquad\textbf{(C) }\text{a right triangle} \qquad$
$\textbf{(D) }\text{a triangle and a trapezoid}\qquad\textbf{(E) }\text{a quadrilateral} $
2021 Saudi Arabia Training Tests, 24
Let $ABC$ be triangle with $M$ is the midpoint of $BC$ and $X, Y$ are excenters with respect to angle $B,C$. Prove that $MX$, $MY$ intersect $AB$, $AC$ at four points that are vertices of circumscribed quadrilateral.
2021 IMO, 1
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
2002 Putnam, 4
An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with uniform probability. Your objective is to select $n$ in an ODD number of guess. After each incorrect guess, you are informed whether $n$ is higher or lower, and you $\textbf{must}$ guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than $\tfrac{2}{3}$.
2007 Romania National Olympiad, 4
Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$.
a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$.
b) Give an example of a non-constant function $f$ with property $(P)$.
c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.
2015 USAMTS Problems, 3
For all positive integers $n$, show that:
$$ \dfrac1n \sum^n _{k=1} \dfrac{k \cdot k! \cdot {n\choose k}}{n^k} = 1$$
2014 Silk Road, 2
Let $w$ be the circumcircle of non-isosceles acute triangle $ABC$. Tangent lines to $w$ in $A$ and $B$ intersect at point $S$. Let M be the midpoint of $AB$, and $H$ be the orthocenter of triangle $ABC$. The line $HA$ intersects lines $CM$ and $CS$ at points $M_a$ and $S_a$, respectively. The points $M_b$ and $S_b$ are defined analogously. Prove that $M_aS_b$ and $M_bS_a$ are the altitudes of triangle $M_aM_bH$.
2010 Lithuania National Olympiad, 4
Arrange arbitrarily $1,2,\ldots ,25$ on a circumference. We consider all $25$ sums obtained by adding $5$ consecutive numbers. If the number of distinct residues of those sums modulo $5$ is $d$ $(0\le d\le 5)$,find all possible values of $d$.
2019 Thailand TSTST, 1
Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.
2022 Durer Math Competition Finals, 5
Benedek draws circles with the same center in the following way. The first circle he draws has radius $1$. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram).
What is the smallest $n$ fow which the radius of the $n$-th circle is an integer greater than $1$?
[img]https://cdn.artofproblemsolving.com/attachments/e/2/afa6d5ead6f2252aa821028370a3768912e674.png[/img]
LMT Speed Rounds, 2022 F
[b]p1.[/b] Each box represents $1$ square unit. Find the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/0/0/f8f8ad6d771f3bbbc59b374a309017cecdce5a.png[/img]
[b]p2.[/b] Evaluate $(3^3)\sqrt{5^2-2^4} -5 \cdot 9$.
[b]p3.[/b] Find the last two digits of $21^3$.
[b]p4.[/b] Let $L$, $M$, and $T$ be distinct prime numbers. Find the least possible odd value of$ L+M +T$ .
[b]p5.[/b]Two circles have areas that sum to $20\pi$ and diameters that sum to $12$. Find the radius of the smaller circle.
[b]p6.[/b] Zach and Evin each independently choose a date in the year $2022$, uniformly and randomly. The probability that at least one of the chosen dates is December $17$, $2022$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $A$.
[b]p7.[/b] Let $L$ be a list of $2023$ real numbers with medianm. When any two numbers are removed from $L$, its median is still $m$. Find the greatest possible number of distinct values in $L$.
[b]p8.[/b] Some children and adults are eating a delicious pile of sand. Children comprise $20\%$ of the group and combined, they consume $80\%$ of the sand. Given that on average, each child consumes $N$ pounds of sand and on average, each adult consumes $M$ pounds of sand, find $\frac{N}{M}$.
[b]p9.[/b] An integer $N$ is chosen uniformly and randomly from the set of positive integers less than $100$. The expectedm number of digits in the base-$10$-representation of $N$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p10.[/b] Dunan is taking a calculus course in which the final exam counts for $15\%$ of the total grade. Dunan wishes to have an $A$ in the course, which is defined as a grade of $93\%$ or above. When counting everything but the final exam, he currently has a $92\%$ in the course. What is the minimum integer grade Dunan must get on the final exam in order to get an $A$ in the course?
[b]p11.[/b] Norbert, Eorbert, Sorbert, andWorbert start at the origin of the Cartesian Plane and walk in the positive $y$, positive $x$, negative $y$, and negative $x$ directions respectively at speeds of $1$, $2$, $3$, and $4$ units per second respectively. After how many seconds will the quadrilateral with a vertex at each person’s location have area $300$?
[b]p12.[/b] Find the sum of the unique prime factors of $1020201$.
[b]p13.[/b] HacoobaMatata rewrites the base-$10$ integers from $0$ to $30$ inclusive in base $3$. How many times does he write the digit $1$?
[b]p14.[/b] The fractional part of $x$ is $\frac17$. The greatest possible fractional part of $x^2$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p15.[/b] For howmany integers $x$ is $-2x^2 +8 \ge x^2 -3x +2$?
[b]p16.[/b] In the figure below, circle $\omega$ is inscribed in square $EFGH$, which is inscribed in unit square $ABCD$ such that $\overline{EB} = 2\overline{AE}$. If the minimum distance from a point on $\omega$ to $ABCD$ can be written as $\frac{P-\sqrt{Q}}{R}$ with $Q$ square-free, find $10000P +100Q +R$.
[img]https://cdn.artofproblemsolving.com/attachments/a/1/c6e5400bc508ab14f34987c9f5f4039daaa4d6.png[/img]
[b]p17.[/b] There are two base number systems in use in the LHS Math Team. One member writes “$13$ people usemy base, while $23$ people use the other, base $12$.” Another member writes “out of the $34$ people in the club, $10$ use both bases while $9$ use neither.” Find the sum of all possible numbers ofMath Team members, as a regular decimal number.
[b]p18.[/b] Sam is taking a test with $100$ problems. On this test the questions gradually get harder in such a way that for question $i$ , Sam has a $\frac{(101-i)^2}{ 100} \%$ chance to get the question correct. Suppose the expected number of questions Sam gets correct can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p19.[/b] In an ordered $25$-tuple, each component is an integer chosen uniformly and randomly from $\{1,2,3,4,5\}$. Ephram and Zach both copy this tuple into a $5\times 5$ grid, both starting from the top-left corner. Ephram writes five components from left to right to fill one row before continuing down to the next row. Zach writes five components from top to bottom to fill one column before continuing right to the next column. Find the expected number of spaces on their grids where Zach and Ephram have the same integer written.
[b]p20.[/b] In $\vartriangle ABC$ with circumcenter $O$ and circumradius $8$, $BC = 10$. Let $r$ be the radius of the circle that passes through $O$ and is tangent to $BC$ at $C$. The value of $r^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $1000m+n$.
[b]p21.[/b] Find the number of integer values of $n$ between $1$ and $100$ inclusive such that the sum of the positive divisors of $2n$ is at least $220\%$ of the sum of the divisors of $n$.
[b]p22.[/b] Twenty urns containing one ball each are arranged in a circle. Ernie then moves each ball either $1$, $2$ or $3$ urns clockwise, chosen independently, uniformly, and randomly. The expected number of empty urns after this process is complete can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p23.[/b] Hannah the cat begins at $0$ on a number line. Every second, Hannah jumps $1$ unit in the positive or negative direction, chosen uniformly at random. After $7$ seconds,Hannah‘s expected distance from $0$, in units, can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$.
[b]p24.[/b] Find the product of all primes $p < 30$ for which there exists an integer $n$ such that $p$ divides $n +(n +1)^{-1}\,\, (mod \,\,p)$.
[b]p25.[/b] In quadrilateral $ABCD$, $\angle ABD = \angle CBD = \angle C AD$, $AB = 9$, $BC = 6$, and $AC = 10$. The area of $ABCD$ can be expressed as $\frac{P\sqrt{Q}}{R}$ with $Q$ squarefree and $P$ and $R$ relatively prime. Find $10000P +100Q +R$.
[img]https://cdn.artofproblemsolving.com/attachments/4/8/28569605b262c8f26e685e27f5f261c70a396c.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 Brazil National Olympiad, 2
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.
[b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal.
[b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.
2009 Regional Olympiad of Mexico Center Zone, 4
Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.
2010 AMC 8, 11
The top of one tree is $16$ feet higher than the top of another tree. The height of the $2$ trees are at a ratio of $3:4$. In feet, how tall is the taller tree?
$ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 64 \qquad\textbf{(C)}\ 80 \qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 112 $
2025 Kosovo National Mathematical Olympiad`, P2
Find all real numbers $a$ and $b$ that satisfy the system of equations:
$$\begin{cases}
a &= \frac{2}{a+b} \\
\\
b &= \frac{2}{3a-b} \\
\end{cases}$$
2016 ISI Entrance Examination, 7
$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of
$$\int_0^1(x-f(x))^{2016}dx$$
Cono Sur Shortlist - geometry, 1993.8
In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.
2019 Middle European Mathematical Olympiad, 8
Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$.
[i]Proposed by Alain Rossier, Switzerland[/i]
2016 India Regional Mathematical Olympiad, 6
Show that the infinite arithmetic progression $\{1,4,7,10 \ldots\}$ has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples $\{a_1, a_2 , a_3 \}$ and $\{b_1, b_2 ,b_3\}$ in harmonic progression , one has $$\frac{a_1} {b_1} \ne \frac {a_2}{b_2}$$.
2011 Balkan MO Shortlist, A1
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
1978 USAMO, 3
An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.
2014 Turkey Team Selection Test, 3
Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds.
Prove that
\[ED=AC-AB \iff R=2r+r_a.\]
1977 AMC 12/AHSME, 7
If $t = \frac{1}{1 - \sqrt[4]{2}}$, then $t$ equals
\[ \text{(A)}\ (1 - \sqrt[4]{2})(2 - \sqrt{2}) \qquad \text{(B)}\ (1 - \sqrt[4]{2})(1 + \sqrt{2}) \qquad \text{(C)}\ (1 + \sqrt[4]{2})(1 - \sqrt{2}) \]
\[ \text{(D)}\ (1 + \sqrt[4]{2})(1 + \sqrt{2}) \qquad \text{(E)} -(1 + \sqrt[4]{2})(1 + \sqrt{2}) \]
2014 Kyiv Mathematical Festival, 1a
a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges?
b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges?
[size=85](Kyiv mathematical festival 2014)[/size]