Found problems: 85335
1997 AMC 12/AHSME, 9
In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
[asy]defaultpen(linewidth(.8pt));
dotfactor=4;
pair A = (0,2);
pair B = origin;
pair C = (2,0);
pair D = (2,2);
pair E = midpoint(A--D);
pair F = foot(C,B,E);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW);
draw(A--B--C--D--cycle);
draw(B--E);
draw(C--F);
draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$
2009 District Olympiad, 1
Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.
2018 CCA Math Bonanza, L4.4
Alice and Billy are playing a game on a number line. They both start at $0$. Each turn, Alice has a $\frac{1}{2}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{2}$ chance of moving $1$ unit in the negative direction, while Billy has a $\frac{2}{3}$ chance of moving $1$ unit in the positive direction, and a $\frac{1}{3}$ chance of moving $1$ unit in the negative direction. Alice and Billy alternate turns, with Alice going first. If a player reaches $2$, they win and the game ends, but if they reach $-2$, they lose and the other player wins, and the game ends. What is the probability that Billy wins?
[i]2018 CCA Math Bonanza Lightning Round #4.4[/i]
2013 239 Open Mathematical Olympiad, 4
For positive numbers $a, b, c$ satisfying condition $a+b+c<2$, Prove that
$$ \sqrt{a^2 +bc}+\sqrt{b^2 +ca}+\sqrt{c^2 + ab}<3. $$
1970 IMO Longlists, 30
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that
\[1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .\]
2016 Argentina National Olympiad, 6
Let $AB$ be a segment of length $1$. Several particles start moving simultaneously at constant speeds from $A$ up to$ B$. As soon as a particle reaches $B$ , turns around and goes to $A$; when it reaches $A$, begins to move again towards $ B$ , and so on indefinitely.
Find all rational numbers$ r>1$ such that there exists an instant $t$ with the following property:
For each $n\ge 1$ , if $n+1$ particles with constant speeds $1,r,r^2,…,r^n$ move as described, at instant $t$, they all lie at the same interior point of segment $AB$.
2013 Sharygin Geometry Olympiad, 2
Let $ABCD$ is a tangential quadrilateral such that $AB=CD>BC$. $AC$ meets $BD$ at $L$. Prove that $\widehat{ALB}$ is acute.
[hide]According to the jury, they want to propose a more generalized problem is to prove $(AB-CD)^2 < (AD-BC)^2$, but this problem has appeared some time ago[/hide]
2021 Durer Math Competition Finals, 4
Indians find those sequences of non-negative real numbers $x_0, x_1,...$ [i]mystical [/i]t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?
2018 Harvard-MIT Mathematics Tournament, 2
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?
2015 ISI Entrance Examination, 6
Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$
2002 IMO, 6
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]
2003 Poland - Second Round, 4
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.
2022 Rioplatense Mathematical Olympiad, 5
The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=P(ABCD)+32$. Determine the length of the side $CD$.
1998 AIME Problems, 5
Given that $A_k=\frac{k(k-1)}2\cos\frac{k(k-1)\pi}2,$ find $|A_{19}+A_{20}+\cdots+A_{98}|.$
2014 Puerto Rico Team Selection Test, 3
Is it possible to tile an $8\times8$ board with dominoes ($2\times1$ tiles) so that no two dominoes form a $2\times2$ square?
2016 ASDAN Math Tournament, 7
Eddy and Moor play a game with the following rules:
[list=a]
[*] The game begins with a pile of $N$ stones, where $N$ is a positive integer. [/*]
[*] The $2$ players alternate taking turns (e.g. Eddy moves, then Moor moves, then Eddy moves, and so on). [/*]
[*] During a player's turn, given $a$ stones remaining in the pile, the player may remove $b$ stones from the pile, where $\gcd(a,b)=1$ and $b\leq a$. [/*]
[*] If a player cannot make a move, they lose. [/*]
[/list]
For example, if Eddy goes first and $N=4$, then Eddy can remove $3$ stones from the pile (since $3\leq4$ and $\gcd(3,4)=1$), leaving $1$ stone in the pile. Moor can then remove $1$ stone from the pile (since $1\leq1$ and $\gcd(1,1)=1$), leaving $0$ stones in the pile. Since Eddy cannot remove stones from an empty pile, he cannot make a move, and therefore loses.
Both Eddy and Moor want to win, so they will both always make the best possible move. If Eddy moves first, for how many values of $N<2016$ can Eddy win no matter what moves Moor chooses?
2008 HMNT, 10
Find the largest positive integer $n$ such that $n^3 + 4n^2 - 15n - 18$ is the cube of an integer.
1974 Putnam, B5
Show that
$$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$
for every integer $n\geq 0.$
1970 IMO Longlists, 23
Let $E$ be a finite set, $P_E$ the family of its subsets, and $f$ a mapping from $P_E$ to the set of non-negative reals, such that for any two disjoint subsets $A,B$ of $E$, $f(A\cup B)=f(A)+f(B)$. Prove that there exists a subset $F$ of $E$ such that if with each $A \subset E$, we associate a subset $A'$ consisting of elements of $A$ that are not in $F$, then $f(A)=f(A')$ and $f(A)$ is zero if and only if $A$ is a subset of $F$.
2010 Romania Team Selection Test, 1
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
2018 Junior Balkan MO, 1
Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.
2023 SG Originals, Q4
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
2019 Saudi Arabia JBMO TST, 4
A positive integer $n$ is called $nice$, if the sum of the squares of all its positive divisors is equal to $(n+3)^2$. Prove that if $n=pq$ is nice, where $p, q$ are not necessarily distinct primes, then $n+2$ and $2(n+1)$ are simultaneously perfect squares.
2016 Thailand TSTST, 4
Let $a, b, c$ be positive reals such that $4(a+b+c)\geq\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Define
\begin{align*}
&A =\sqrt{\frac{3a}{a+2\sqrt{bc}}}+\sqrt{\frac{3b}{b+2\sqrt{ca}}}+\sqrt{\frac{3c}{c+2\sqrt{ab}}} \\
&B =\sqrt{a}+\sqrt{b}+\sqrt{c} \\
&C =\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}.
\end{align*}
Prove that $$A\leq 2B\leq 4C.$$
MOAA Gunga Bowls, 2019
[u]Set 1[/u]
[b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day?
[b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$?
[b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats?
[u]Set 2[/u]
[b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime?
[b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$.
[b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$.
[u]Set 3[/u]
[b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ .
[b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ .
[b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$?
[u]Set 4[/u]
[b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$.
[b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings.
[b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$.
[u]Set 5[/u]
[b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$.
[b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins?
[b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$.
PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].