Found problems: 85335
2024 Malaysian IMO Training Camp, 4
Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$
[i]Proposed by Wong Jer Ren[/i]
1954 AMC 12/AHSME, 9
A point $ P$ is outside a circle and is $ 13$ inches from the center. A secant from $ P$ cuts the circle at $ Q$ and $ R$ so that the external segment of the secant $ PQ$ is $ 9$ inches and $ QR$ is $ 7$ inches. The radius of the circle is:
$ \textbf{(A)}\ 3" \qquad \textbf{(B)}\ 4" \qquad \textbf{(C)}\ 5" \qquad \textbf{(D)}\ 6" \qquad \textbf{(E)}\ 7"$
2023 AMC 12/AHSME, 17
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$?
$\textbf{(A)}\ 12\sqrt 3 \qquad\textbf{(B)}\ 8\sqrt 6 \qquad\textbf{(C)}\ 14\sqrt 2 \qquad\textbf{(D)}\ 20\sqrt 2 \qquad\textbf{(E)}\ 15\sqrt 3$
Novosibirsk Oral Geo Oly IX, 2016.6
An arbitrary point $M$ inside an equilateral triangle $ABC$ was connected to vertices. Prove that on each side the triangle can be selected one point at a time so that the distances between them would be equal to $AM, BM, CM$.
2020 BMT Fall, Tie 2
Quadrilateral $ABCD$ is cyclic with $AB = CD = 6$. Given that $AC = BD = 8$ and $AD+3 = BC$, the area of $ABCD$ can be written in the form $\frac{p\sqrt{q}}{r}$, where $p, q$, and $ r$ are positive integers such that $p$ and $ r$ are relatively prime and that $q$ is square-free. Compute $p + q + r$.
Gheorghe Țițeica 2024, P2
Find all monotonic and twice differentiable functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f''+4f+3f^2+8f^3=0.$$
1995 AMC 12/AHSME, 24
There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that \[A \log_{200} 5 + B \log_{200} 2 = C. \] What is $A+B+C$?
$\textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
2010 Today's Calculation Of Integral, 595
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$
2009 Kumamoto University entrance exam/Medicine
2018 CMIMC CS, 2
Consider the natural implementation of computing Fibonacci numbers:
\begin{tabular}{l}
1: \textbf{FUNCTION} $\text{FIB}(n)$: \\
2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\
3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$
\end{tabular}
When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur?
2018 Malaysia National Olympiad, A4
Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.
Ukraine Correspondence MO - geometry, 2015.11
Let $ABC$ be an non- isosceles triangle, $H_a$, $H_b$, and $H_c$ be the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively, and $M_a$, $M_b$, and $M_c$ be the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The circumscribed circles of triangles $AH_bH_c$ and $AM_bM_c$ intersect for second time at point $A'$. The circumscribed circles of triangles $BH_cH_a$ and $BM_cM_a$ intersect for second time at point $B'$. The circumscribed circles of triangles $CH_aH_b$ and $CM_aM_b$ intersect for second time at point $C'$. Prove that points $A', B'$ and $C'$ lie on the same line.
2017 Romania National Olympiad, 3
In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
2018 Germany Team Selection Test, 3
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
2023 Malaysia IMONST 2, 2
Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$. How many cats of each breed does he have?
1987 AMC 12/AHSME, 15
If $(x, y)$ is a solution to the system
\[ xy=6 \qquad \text{and} \qquad x^2y+xy^2+x+y=63, \]
find $x^2+y^2.$
$ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ \frac{1173}{32} \qquad\textbf{(C)}\ 55 \qquad\textbf{(D)}\ 69 \qquad\textbf{(E)}\ 81 $
2017 ASDAN Math Tournament, 10
Alice lives on a continent with $6$ countries labeled $1$ through $6$. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county $1$, what is the expected number of countries that she can reach (including country $1$)?
2019 International Zhautykov OIympiad, 2
Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality :
$\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$
2019 Taiwan APMO Preliminary Test, P6
Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$
(1)Find $f(0)$
(2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$
2007 Korea Junior Math Olympiad, 8
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$).
Prove that there are infi nite number of [i]Primes of the Year[/i].
2016 Postal Coaching, 2
Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$
$$f(xf(y) - yf(x)) = f(xy) - xy.$$
2013 Singapore MO Open, 1
Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$-th powers of the digits of $a_n$. Do there exist distinct positive integers $i$, $j$ such that $a_i=a_j$?
1971 Kurschak Competition, 3
There are $30$ boxes each with a unique key. The keys are randomly arranged in the boxes, so that each box contains just one key and the boxes are locked. Two boxes are broken open, thus releasing two keys. What is the probability that the remaining boxes can be opened without forcing them?
2018 China Team Selection Test, 4
Let $p$ be a prime and $k$ be a positive integer. Set $S$ contains all positive integers $a$ satisfying $1\le a \le p-1$, and there exists positive integer $x$ such that $x^k\equiv a \pmod p$.
Suppose that $3\le |S| \le p-2$. Prove that the elements of $S$, when arranged in increasing order, does not form an arithmetic progression.
2019 IFYM, Sozopol, 1
A football tournament is played between 5 teams, each two of which playing exactly one match. 5 points are awarded for a victory and 0 – for a loss. In case of a draw 1 point is awarded to both teams, if no goals are scored, and 2 – if they have scored any. In the final ranking the five teams had points that were 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.