Found problems: 85335
2008 Mexico National Olympiad, 1
A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out $1,2,3,1,2,3,\dots$ around the circle (that is, clockwise, and each person says a number). The people who say $2$ or $3$ immediately lose, and this continues until the last knight is left, the winner.
Numbering the knights initially as $1,2,\dots,n$, find all values of $n$ such that knight $2008$ is the winner.
2003 Alexandru Myller, 3
Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit.
[i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]
2008 National Olympiad First Round, 33
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 66
\qquad\textbf{(C)}\ 68
\qquad\textbf{(D)}\ 70
\qquad\textbf{(E)}\ 72
$
2000 Putnam, 4
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.
1998 Romania Team Selection Test, 2
A parallelepiped has surface area 216 and volume 216. Show that it is a cube.
2014 South East Mathematical Olympiad, 4
Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]
1999 National Olympiad First Round, 5
Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$. $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$. $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and $ \left[AC\right]$, respectively. If $ M$ and $ N$ are the foot of perpendiculars from $ S$ and $ R$ to $ PT$, then find $ \left|MN\right|$.
$\textbf{(A)}\ \frac {9\sqrt {13} }{26} \qquad\textbf{(B)}\ \frac {12 \minus{} 2\sqrt {13} }{13} \qquad\textbf{(C)}\ \frac {5\sqrt {13} \plus{} 20}{13} \qquad\textbf{(D)}\ 15\sqrt {3} \qquad\textbf{(E)}\ \frac {10\sqrt {13} }{13}$
1998 All-Russian Olympiad, 5
We are given five watches which can be winded forward. What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?
2023 Harvard-MIT Mathematics Tournament, 4
Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.
2024 Vietnam National Olympiad, 1
For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$.
A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$.
b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$
2022 CMIMC, 11
Let $\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0}$ be two infinite sequences of real numbers, such that $\varepsilon_i \in \{-1,1\}$ for all $i$, and the numbers $\theta_i$ obey$$\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1$$and $\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}$. Compute the sum of all possible values of $$\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)$$
[i]Proposed by Grant Yu[/i]
2003 AMC 8, 11
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40$ dollars on Thursday?
$\textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 39.60 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 40.40 \qquad
\textbf{(E)}\ 44$
May Olympiad L2 - geometry, 2001.4
Ten coins of $1$ cm radius are placed around a circle as indicated in the figure.
Each coin is tangent to the circle and its two neighboring coins.
Prove that the sum of the areas of the ten coins is twice the area of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/edf7a7d39d749748f4ae818853cb3f8b2b35b5.gif[/img]
2021 CMIMC Integration Bee, 3
$$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\sin(2x)\,dx$$
[i]Proposed by Connor Gordon[/i]
2025 PErA, P4
Let \( ABC \) be an acute-angled scalene triangle. Let \( B_1 \) and \( B_2 \) be points on the rays \( BC \) and \( BA \), respectively, such that \( BB_1 = BB_2 = AC \). Similarly, let \( C_1 \) and \( C_2 \) be points on the rays \( CB \) and \( CA \), respectively, such that \( CC_1 = CC_2 = AB \). Prove that if \( B_1B_2 \) and \( C_1C_2 \) intersect at \( K \), then \( AK \) is parallel to \( BC \).
1953 AMC 12/AHSME, 1
A boy buys oranges at $ 3$ for $ 10$ cents. He will sell them at $ 5$ for $ 20$ cents. In order to make a profit of $ \$ 1.00$, he must sell:
$ \textbf{(A)}\ 67 \text{ oranges} \qquad\textbf{(B)}\ 150 \text{ oranges} \qquad\textbf{(C)}\ 200 \text{ oranges} \\
\textbf{(D)}\ \text{an infinite number of oranges} \qquad\textbf{(E)}\ \text{none of these}$
2020 Estonia Team Selection Test, 1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
2023 IMAR Test, P1
Let $ABC$ be an acute triangle, and let $D,E,F$ be the feet of its altitudes from $A,B,C$ respectively. The lines $AB{}$ and $DE$ cross at $K{}$ and the lines $AC$ and $DF$ cross at $L{}.$ Let $M$ be the midpoint of the side $BC$ and let the line $AM$ cross the circle $(ABC)$ again at $N{}.$ The parallel through $M{}$ to $EF$ crosses the line $KL$ at $P{}.$ Prove that the triangle $MNP$ is isosceles.
MathLinks Contest 5th, 3.2
Let $0 < a_1 < a_2 <... < a_{16} < 122$ be $16$ integers. Prove that there exist integers $(p, q, r, s)$, with $1 \le p < r \le s < q \le 16$, such that $a_p + a_q = a_r + a_s$.
An additional $2$ points will be awarded for this problem, if you can find a larger bound than $122$ (with proof).
2013 China National Olympiad, 3
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
1993 Dutch Mathematical Olympiad, 5
Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.
2017 Bulgaria EGMO TST, 3
Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.
2024-25 IOQM India, 29
Let $n = 2^{19}3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$.What is the number formed by taking the last two digits of $M$ (in the same order)?
2024 CMIMC Team, 4
Eric and Christina are playing a game with $n$ stones. They alternate taking some number of stones from the pile, with Eric going first. The number of stones Eric takes from the pile must be a power of $3$ (e.g. 1, 3, 9, 27, ...), while the number of stones Christina takes must be a power of $2$ (e.g. 1, 2, 4, 8, ...). Whoever takes the last stone wins. Find the sum of all $1\leq n \leq 100$ for which Eric has a winning strategy.
[i]Proposed by Connor Gordon[/i]
Brazil L2 Finals (OBM) - geometry, 2013.5
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry?