Found problems: 85335
2016 LMT, 1
Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by Matthew Weiss
2005 Korea - Final Round, 6
A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$
1967 IMO Longlists, 30
Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.
2022 Belarusian National Olympiad, 11.7
Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$.
Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$
1955 Miklós Schweitzer, 9
[b]9.[/b] Show that to any elliptic paraboloid $\varphi_1$ there may be found an elliptic paraboloid $\varphi_2$ (other than $\varphi_1$) and an affinity $\phi$ which maps $\varphi_1$ onto $\varphi_2$ and has the following property: If $P$ is any point of $\varphi_1$ such that $\phi(P) \neq P$, then the straight line connecting $P$ and $\phi(P)$ is a common tangent of the two paraboloids. [b](G. 18)[/b]
Swiss NMO - geometry, 2005.8
Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.
2007 National Olympiad First Round, 25
Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$?
$
\textbf{(A)}\ \dfrac 12
\qquad\textbf{(B)}\ \dfrac {\sqrt 3}2
\qquad\textbf{(C)}\ \dfrac {\sqrt 2}2
\qquad\textbf{(D)}\ \sqrt 3 -1
\qquad\textbf{(E)}\ \text{None of the above}
$
2022 Czech-Austrian-Polish-Slovak Match, 2
Find all functions $f: \mathbb{R^{+}} \rightarrow \mathbb {R^{+}}$ such that $f(f(x)+\frac{y+1}{f(y)})=\frac{1}{f(y)}+x+1$ for all $x, y>0$.
[i]Proposed by Dominik Burek, Poland[/i]
2022 Mexican Girls' Contest, 4
Let $k$ be a positive integer and $m$ be an odd integer. Prove that there exists a positive integer $n$ such that $n^n-m$ is divisible by $2^k$.
ABMC Accuracy Rounds, 2021
[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers
$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit?
[b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice.
[b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$.
[b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side.
[b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number?
[b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score.
[b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively
prime positive integers $a$, $b$, Find $a + b$.
[b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$.
[b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$.
[b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.
[b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1991 Swedish Mathematical Competition, 5
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
2014 Contests, 2
Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle [i]gergonnians[/i].
(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.
(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.
1969 Putnam, B3
The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and
$$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$
Show that $ \pi T_{1}^{2}=2.$
KoMaL A Problems 2023/2024, A. 865
A crossword is a grid of black and white cells such that every white cell belongs to some $2\times 2$ square of white cells. A word in the crossword is a contiguous sequence of two or more white cells in the same row or column, delimited on each side by either a black cell or the boundary of the grid.
Show that the total number of words in an $n\times n$ crossword cannot exceed $(n+1)^2/2$.
[i]Proposed by Nikolai Beluhov, Bulgaria[/i]
2005 Harvard-MIT Mathematics Tournament, 7
Two ants, one starting at $ (-1, 1) $, the other at $ (1, 1) $, walk to the right along the parabola $ y = x^2 $ such that their midpoint moves along the line $ y = 1 $ with constant speed $1$. When the left ant first hits the line $ y = \frac {1}{2} $, what is its speed?
2005 National Olympiad First Round, 31
If the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real roots, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, at most how many distinct real roots can the equation $f(x)g(x) = 0$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$
2015 NIMO Problems, 2
Consider the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \in \{-1, 0, 1\}$ and $(x,y) \neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints?
[i]Proposed by Evan Chen[/i]
1989 Bulgaria National Olympiad, Problem 3
Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that:
(a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$;
(b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.
1989 IMO Longlists, 75
Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$
2023 HMNT, 6
There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.
2011 Postal Coaching, 4
Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?
2021 CCA Math Bonanza, I11
An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\ \ \ \text{and} \ \ \ y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\]
the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]2021 CCA Math Bonanza Individual Round #11[/i]
2014 ASDAN Math Tournament, 1
Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.
2011 Today's Calculation Of Integral, 690
Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.
2004 All-Russian Olympiad Regional Round, 11.6
Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?