Found problems: 85335
CIME I 2018, 1
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
2022 ELMO Revenge, 1
Let $ABC$ and $DBC$ be triangles with incircles touching at a point $P$ on $BC.$ Points $A,D$ lie on the same side of $BC$ and $DB < AB < DC < AC.$ The bisector of $\angle BDC$ meets line $AP$ at $X,$ and the altitude from $A$ meets $DP$ at $Y.$ Point $Z$ lies on line $XY$ so $ZP \perp BC.$ Show the reflection of $A$ over $BC$ is on line $ZD.$
[i]Proposed by squareman (Evan Chang), USA[/i]
2012 AMC 12/AHSME, 1
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
1949 Miklós Schweitzer, 8
The four sides of a skew quadrangle and the two segments joining the midpoints of the opposite sides are realized by rigid bars. The bars are linked by hinges. Prove that this apparatus is not rigid.
P.S: The 1949 Miklos Schweitzer competition had only 8 problems!
2005 China Team Selection Test, 2
Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying :
For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.
2004 Kazakhstan National Olympiad, 2
A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?
2010 JBMO Shortlist, 1
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.
MBMT Guts Rounds, 2015.22
In rhombus ABCD, $\angle A = 60^\circ$. Rhombus $BEFG$ is constructed, where $E$ and $G$ are the midpoints of $BC$ and $AB$, respectively. Rhombus $BHIJ$ is constructed, where $H$ and $J$ are the midpoints of $BE$ and $BG$, respectively. This process is repeated forever. If the area of $ABCD$ and the sum of the areas of all of the rhombi are both integers, compute the smallest possible value of $AB$.
2012 JBMO ShortLists, 3
Decipher the equality :
\[(\overline{VER}-\overline{IA})=G^{R^E} (\overline {GRE}+\overline{ECE}) \]
assuming that the number $\overline {GREECE}$ has a maximum value .Each letter corresponds to a unique digit from $0$ to $9$ and different letters correspond to different digits . It's also supposed that all the letters $G$ ,$E$ ,$V$ and $I$ are different from $0$.
2025 USAMO, 2
Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.
2018 Switzerland - Final Round, 2
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.
2025 NCMO, 3
Let $\mathcal{S}$ be a set of points in the plane such that for each subset $\mathcal{T}$ of $\mathcal{S}$, there exists a convex $2025$-gon which contains all of the points in $\mathcal{T}$ and none of the rest of the points in $\mathcal{S}$ but not $\mathcal{T}$. Determine the greatest possible number of points in $\mathcal{S}$.
[i]Jason Lee[/i]
2017 Rioplatense Mathematical Olympiad, Level 3, 3
Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that
$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.
2014 IFYM, Sozopol, 2
We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that
$2a_n-1$ is a perfect square.
1996 Spain Mathematical Olympiad, 5
At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.
2016 BMT Spring, 9
Given right triangle $ABC$ with right angle at $C$, construct three external squares $ABDE$, $BCFG$, and $ACHI$. If $DG = 19$ and $EI = 22$, compute the length of $FH$.
MOAA Individual Speed General Rounds, 2018 Ind
[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$.
[b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s?
[b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ .
[b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day?
[b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
[b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$.
[b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces?
[b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$.
[b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance?
[img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img]
[b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$?
[b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$?
[b]p12.[/b] What is the largest prime number that is a factor of $160,401$?
[b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$?
[b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$.
[b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$.
[b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ .
[b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold:
$\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$,
$\bullet$ $k_1 + k_2 + ...+ k_n = 0$.
Determine the number of spicy integers less than $10^6$.
[b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$
$$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system.
[b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not.
[b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 BMT Spring, 10
Find the number of ordered integer triplets $ x, y, z $ with absolute value less than or equal to 100 such that $ 2x^2 + 3y^2 + 3z^2 + 2xy + 2xz - 4yz < 5 $.
2018 Iran MO (2nd Round), 3
Let $a>k$ be natural numbers and $r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n$ be sequences of natural numbers such that:
$(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)$
Prove that these sequences are equal.
1998 Romania National Olympiad, 3
A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that:
a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$.
b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.
PEN F Problems, 4
Suppose that $\tan \alpha =\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove the number $\tan \beta$ for which $\tan 2\beta =\tan 3\alpha$ is rational only when $p^2 +q^2$ is the square of an integer.
2023 Durer Math Competition Finals, 5
For an acute triangle $ABC$, let $O$ be its circumcenter, and let $O_A,O_B,O_C$ be the circumcenter of $BCO,CAO,ABO$ respectively. Show that $AO_A,BO_B,CO_C$ are concurrent.
1998 AIME Problems, 1
For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6, 8^8,$ and $k$?
1999 Baltic Way, 13
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.
2010 Contests, 3
On a line $l$ there are three different points $A$, $B$ and $P$ in that order. Let $a$ be the line through $A$ perpendicular to $l$, and let $b$ be the line through $B$ perpendicular to $l$. A line through $P$, not coinciding with $l$, intersects $a$ in $Q$ and $b$ in $R$. The line through $A$ perpendicular to $BQ$ intersects $BQ$ in $L$ and $BR$ in $T$. The line through $B$ perpendicular to $AR$ intersects $AR$ in $K$ and $AQ$ in $S$.
(a) Prove that $P$, $T$, $S$ are collinear.
(b) Prove that $P$, $K$, $L$ are collinear.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 3)[/i]