Found problems: 85335
1997 AMC 8, 25
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$
2024 IFYM, Sozopol, 6
The positive integers \( a \), \( b \), \( c \), \( d \) are such that \( (a+c)(b+d) = (ab-cd)^2 \). Prove that \( 4ad + 1 \) and \( 4bc + 1 \) are perfect squares of natural numbers.
2011 Iran MO (3rd Round), 2
prove that the number of permutations such that the order of each element is a multiple of $d$ is $\frac{n!}{(\frac{n}{d})!d^{\frac{n}{d}}} \prod_{i=0}^{\frac{n}{d}-1} (id+1)$.
[i]proposed by Mohammad Mansouri[/i]
2006 AMC 8, 5
Points $ A, B, C$ and $ D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?
[asy]size(100);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle,linewidth(1));
draw((0,1)--(1,2)--(2,1)--(1,0)--cycle);
label("$A$", (1,2), N);
label("$B$", (2,1), E);
label("$C$", (1,0), S);
label("$D$", (0,1), W);[/asy]
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 40$
2006 IMC, 1
Let $V$ be a convex polygon.
(a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles.
(b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.
2010 Today's Calculation Of Integral, 575
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
2018 Iran Team Selection Test, 2
Determine the least real number $k$ such that the inequality
$$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$
holds for all real numbers $a,b,c$.
[i]Proposed by Mohammad Jafari[/i]
2022 USAMTS Problems, 3
A positive integer $N$ is called [i]googolicious[/i] if there are exactly $10^{100}$ positive integers $x$ that satisfy \[\left\lfloor \frac{N}{\left\lfloor \frac{N}{x} \right\rfloor } \right\rfloor = x,\] where $z$ denotes the greatest integer less than $z.$ Find, with proof, all googolicious integers $N.$
1997 AMC 8, 13
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$\textbf{(A)}\ 31\% \qquad \textbf{(B)}\ 32\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 35\% \qquad \textbf{(E)}\ 95\%$
2006 Princeton University Math Competition, 10
If $a_1, ... ,a_{12}$ are twelve nonzero integers such that $a^6_1+...·+a^6_{12} = 450697$, what is the value of $a^2_1+...+a^2_{12}$?
2013 AMC 8, 3
What is the value of $4 \cdot (-1+2-3+4-5+6-7+\cdots+1000)$?
$\textbf{(A)}\ -10 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 500 \qquad \textbf{(E)}\ 2000$
2011 International Zhautykov Olympiad, 1
Find the maximum number of sets which simultaneously satisfy the following conditions:
[b]i)[/b] any of the sets consists of $4$ elements,
[b]ii)[/b] any two different sets have exactly $2$ common elements,
[b]iii)[/b] no two elements are common to all the sets.
1961 Kurschak Competition, 3
Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$, and $Q'$ on the circle center $O'$). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/bad305631571323a61b097f149a1bb6855cdc5.png[/img]
2023 Turkey Team Selection Test, 8
Initially the equation
$$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
2018 Hanoi Open Mathematics Competitions, 7
Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$.
1) Calculate $u_5$.
2) Calculate $u_{100} + u_{101}$.
1955 Moscow Mathematical Olympiad, 289
Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.
1997 Portugal MO, 2
Consider the cube $ABCDEFGH$ and denote by, respectively, $M$ and $N$ the midpoints of $[AB]$ and $[CD]$. Let $P$ be a point on the line defined by $[AE]$ and $Q$ the point of intersection of the lines defined by $[PM]$ and $[BF]$. Prove that the triangle $[PQN]$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/0/0/57559efbad87903d087c738df279b055b4aefd.png[/img]
2013 Saudi Arabia BMO TST, 5
We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.
2009 JBMO Shortlist, 2
A group of $n > 1$ pirates of different age owned total of $2009$ coins. Initially each pirate (except the youngest one) had one coin more than the next younger.
a) Find all possible values of $n$.
b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If $n = 7$, find the largest possible number of coins a pirate can have after several days.
LMT Guts Rounds, 2020 F20
Cyclic quadrilateral $ABCD$ has $AC=AD=5, CD=6,$ and $AB=BC.$ If the length of $AB$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,c$ are relatively prime positive integers and $b$ is square-fre,e evaluate $a+b+c.$
[i]Proposed by Ada Tsui[/i]
2013 Hanoi Open Mathematics Competitions, 11
The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.
2013 NIMO Problems, 6
Let $ABC$ be a triangle with $AB = 42$, $AC = 39$, $BC = 45$. Let $E$, $F$ be on the sides $\overline{AC}$ and $\overline{AB}$ such that $AF = 21, AE = 13$. Let $\overline{CF}$ and $\overline{BE}$ intersect at $P$, and let ray $AP$ meet $\overline{BC}$ at $D$. Let $O$ denote the circumcenter of $\triangle DEF$, and $R$ its circumradius. Compute $CO^2-R^2$.
[i]Proposed by Yang Liu[/i]
2003 Germany Team Selection Test, 2
Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that:
\[\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.\]
1971 Miklós Schweitzer, 5
Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]
[i]L. Leindler[/i]
2000 Moldova National Olympiad, Problem 5
Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$