Found problems: 85335
2002 USAMTS Problems, 4
The vertices of a cube have coordinates $(0,0,0),(0,0,4),(0,4,0),(0,4,4),(4,0,0)$,$(4,0,4),(4,4,0)$, and $(4,4,4)$. A plane cuts the edges of this cube at the points $(0,2,0),(1,0,0),(1,4,4)$, and two other points. Find the coordinates of the other two points.
2012 Lusophon Mathematical Olympiad, 6
A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure.
a) Show that the triangles $AOB$ and $COD$ have the equal areas.
b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.
2013 Saudi Arabia Pre-TST, 2.2
The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.
2011 Peru MO (ONEM), 1
We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.
1994 All-Russian Olympiad, 4
On a line are given $n$ blue and $n$ red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors.
(O. Musin)
LMT Speed Rounds, 2016.9
An acute triangle has area $84$ and perimeter $42$, with each side being at least $10$ units long. Let $S$ be the set of points that are within $5$ units of some vertex of the triangle. What fraction of the area of $S$ lies outside the triangle?
[i]Proposed by Nathan Ramesh
2020 CMIMC Team, 8
Simplify $$\dbinom{2020}{1010}\dbinom{1010}{1010}+\dbinom{2019}{1010}\dbinom{1011}{1010}+\cdots+\dbinom{1011}{1010}\dbinom{2019}{1010} + \dbinom{1010}{1010}\dbinom{2020}{1010}.$$
2020 Putnam, A6
For a positive integer $N$, let $f_N$ be the function defined by
\[ f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right). \]
Determine the smallest constant $M$ such that $f_N (x)\le M$ for all $N$ and all real $x$.
2011 Harvard-MIT Mathematics Tournament, 3
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning.
1999 Vietnam National Olympiad, 3
Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that
[b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$
[b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$
1998 All-Russian Olympiad Regional Round, 9.6
At the ends of a checkered strip measuring $1 \times 101$ squares there are two chips: on the left is the chip of the first player, on the right is the second. Per turn dares to move his piece in the direction of the opposite edge of the strip by 1, 2, 3 or 4 cells. In this case, you are allowed to jump over opponent's chip, but it is forbidden to place your chip on the same square with her. The first one to reach the opposite edge of the strip wins. Who wins if the game is played correctly: the one who goes first, or him rival?
2011 Morocco National Olympiad, 1
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.
2023 Indonesia Regional, 3
Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$
2024 Princeton University Math Competition, 3
Let $f(x)=x^2-3x+1,$ and let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the $4$ roots of $f(f(x))=x.$ Evaluate $\lfloor 10\alpha_1\rfloor+ $ $\lfloor 10\alpha_2\rfloor$ $+$ $\lfloor 10\alpha_3\rfloor+\lfloor 10\alpha_4\rfloor.$
1988 Tournament Of Towns, (181) 4
There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)
2002 Taiwan National Olympiad, 4
Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.
2007 All-Russian Olympiad Regional Round, 8.5
There are $ 11$ coins, which are indistinguishable by sight. Nevertheless, among them there are $ 10$ geniune coins ( of weight $ 20$ g each) and one counterfeit (of weight $ 21$ g). You have a two-pan scale which is blanced when the weight in the left-hand pan is twice as much as the weight in the right-hand one. Using this scale only, find the false coin by three weighings.
1949 Moscow Mathematical Olympiad, 162
Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.
2024 AMC 12/AHSME, 1
In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
$
\textbf{(A) }2021 \qquad
\textbf{(B) }2022 \qquad
\textbf{(C) }2023 \qquad
\textbf{(D) }2024 \qquad
\textbf{(E) }2025 \qquad
$
2022 Thailand Online MO, 4
There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
2011 ISI B.Math Entrance Exam, 6
Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.
1997 Iran MO (3rd Round), 1
Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.
2011 USAJMO, 4
A [i]word[/i] is defined as any finite string of letters. A word is a [i]palindrome[/i] if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.
2016 Iran Team Selection Test, 1
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2006 AMC 12/AHSME, 11
Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$?
$ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$
$\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$