Found problems: 85335
2016 IFYM, Sozopol, 4
Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that:
$\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.
1976 Spain Mathematical Olympiad, 3
Through a lens that inverts the image we look at the rearview mirror of our car. If it reflects the license plate of the car that follows us, $CS-3965-EN$, draw the image we receive. Also draw the one obtained by permuting previous transformations, that is, reflecting in the mirror the image that the license plate gives the lens. Is the product of both transformations , reflection in the mirror and refraction through the lens, commutative?
2009 Putnam, A5
Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
2012 Switzerland - Final Round, 8
Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.
1954 AMC 12/AHSME, 47
At the midpoint of line segment $ AB$ which is $ p$ units long, a perpendicular $ MR$ is erected with length $ q$ units. An arc is described from $ R$ with a radius equal to $ \frac{1}{2}AB$, meeting $ AB$ at $ T$. Then $ AT$ and $ TB$ are the roots of:
$ \textbf{(A)}\ x^2\plus{}px\plus{}q^2\equal{}0 \\
\textbf{(B)}\ x^2\minus{}px\plus{}q^2\equal{}0 \\
\textbf{(C)}\ x^2\plus{}px\minus{}q^2\equal{}0 \\
\textbf{(D)}\ x^2\minus{}px\minus{}q^2\equal{}0 \\
\textbf{(E)}\ x^2\minus{}px\plus{}q\equal{}0$
2016 LMT, 24
Let $S$ be a set consisting of all positive integers less than or equal to $100$. Let $P$ be a subset of $S$ such that there do not exist two elements $x,y\in P$ such that $x=2y$. Find the maximum possible number of elements of $P$.
[i]Proposed by Nathan Ramesh
2009 China Team Selection Test, 2
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.
2000 National Olympiad First Round, 7
Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to below conversation,
$B: $ I'm a truth teller.
$C: $ $D$ is a truth teller.
$D: $ $B$ and $E$ are not both truth tellers.
$E: $ $A$ and $B$ are truth tellers.
How many truth tellers are there?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{More information is needed}
$
2004 Swedish Mathematical Competition, 2
In one country there are coins of value $1,2,3,4$ or $5$. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has $100$ coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?
2012 QEDMO 11th, 11
Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.
2006 ITAMO, 3
Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.
2013 Bogdan Stan, 2
Let $ \left( a_n \right) ,\left( b_n \right) $ be two sequences of real numbers from the interval $ (-1,1) $ having the property that
$$ \max\left( \left| a_{n+1} -a_n \right| ,\left| b_{n+1} -b_n \right| \right) \le\frac{1}{(n+4)(n+5)} , $$
for any natural number.
Prove that $ \left| a_nb_n -a_1b_1 \right|\le 1/2, $ for any natural number $ n. $
[i]Cristinel Mortici[/i]
1988 IMO Shortlist, 14
For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?
2025 CMIMC Algebra/NT, 10
Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$
2024 All-Russian Olympiad Regional Round, 9.7
There is a circle which is 1 meter in circumference and a point marked on it. Two cockroaches start running in the same direction from the marked point with different speeds. Whenever the fast one would catch up with the slow one, the slow one would instantly turn around and start running in tho other direction with the same speed. Whenever they would meet face-to-face, the fast one would instantly turn around and start running in tho other direction with the same speed. How far from the marked point could their 100th meeting be?
Revenge EL(S)MO 2024, 3
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
2024 LMT Fall, 9
Find the median of the positive divisors of $6^4-1$.
2022 Greece Team Selection Test, 3
Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.
2021 Stanford Mathematics Tournament, R4
[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different.
[b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself?
[b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ?
[img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img]
[b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2000 IMO Shortlist, 1
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
PEN G Problems, 3
Prove that there exist positive integers $ m$ and $ n$ such that
\[ \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.\]
2014 HMNT, 1
What is the smallest positive integer $n$ which cannot be written in any of the following forms?
$\bullet$ $n = 1 + 2 +... + k$ for a positive integer $k$.
$\bullet$ $n = p^k$ for a prime number $p$ and integer $k$.
$\bullet$ $n = p + 1$ for a prime number $p$.
2016 Taiwan TST Round 3, 3
You are responsible for arranging a banquet for an agency. In the agency, some pairs of agents are enemies. A group of agents are called [i]avengers[/i], if and only if the number of agents in the group is odd and at least $3$, and it is possible to arrange all of them around a round table so that every two neighbors are enemies.
You figure out a way to assign all agents to $11$ tables so that any two agents on the same tables are not enemies, and that’s the minimum number of tables you can get. Prove that there are at least $2^{10}-11$ avengers in the agency.
This problem is adapted from 2015 IMO Shortlist C7.
2023 LMT Fall, 16
Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)?
[i]Proposed byMuztaba Syed[/i]
2014 LMT, Individual
[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$?
[b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors?
[b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$?
[b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain?
[b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$)
[b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$.
[b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides?
[b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent?
[b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$.
[b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$?
[b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$?
[b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?
[b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?
[b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$.
[b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/
[b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles?
[b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ .
[b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer?
[b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?
[b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].