Found problems: 85335
2000 Mongolian Mathematical Olympiad, Problem 4
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
1987 Tournament Of Towns, (145) 2
Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .
2014 Germany Team Selection Test, 1
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
2020 CHMMC Winter (2020-21), 9
For a positive integer $m$, let $\varphi(m)$ be the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime, and let $\sigma(m)$ be the sum of the positive divisors of $m$. Find the sum of all even positive integers $n$ such that
\[
\frac{n^5\sigma(n) - 2}{\varphi(n)}
\]
is an integer.
2022 BmMT, Ind. Round
[b]p1.[/b] Nikhil computes the sum of the first $10$ positive integers, starting from $1$. He then divides that sum by 5. What remainder does he get?
[b]p2.[/b] In class, starting at $8:00$, Ava claps her hands once every $4$ minutes, while Ella claps her hands once every $6$ minutes. What is the smallest number of minutes after $8:00$ such that both Ava and Ella clap their hands at the same time?
[b]p3.[/b] A triangle has side lengths $3$, $4$, and $5$. If all of the side lengths of the triangle are doubled, how many times larger is the area?
[b]p4.[/b] There are $50$ students in a room. Every student is wearing either $0$, $1$, or $2$ shoes. An even number of the students are wearing exactly $1$ shoe. Of the remaining students, exactly half of them have $2$ shoes and half of them have $0$ shoes. How many shoes are worn in total by the $50$ students?
[b]p5.[/b] What is the value of $-2 + 4 - 6 + 8 - ... + 8088$?
[b]p6.[/b] Suppose Lauren has $2$ cats and $2$ dogs. If she chooses $2$ of the $4$ pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs?
[b]p7.[/b] Let triangle $\vartriangle ABC$ be equilateral with side length $6$. Points $E$ and $F$ lie on $BC$ such that $E$ is closer to $B$ than it is to $C$ and $F$ is closer to $C$ than it is to $B$. If $BE = EF = FC$, what is the area of triangle $\vartriangle AFE$?
[b]p8.[/b] The two equations $x^2 + ax - 4 = 0$ and $x^2 - 4x + a = 0$ share exactly one common solution for $x$. Compute the value of $a$.
[b]p9.[/b] At Shreymart, Shreyas sells apples at a price $c$. A customer who buys $n$ apples pays $nc$ dollars, rounded to the nearest integer, where we always round up if the cost ends in $.5$. For example, if the cost of the apples is $4.2$ dollars, a customer pays $4$ dollars. Similarly, if the cost of the apples is $4.5$ dollars, a customer pays $5$ dollars. If Justin buys $7$ apples for $3$ dollars and $4$ apples for $1$ dollar, how many dollars should he pay for $20$ apples?
[b]p10.[/b] In triangle $\vartriangle ABC$, the angle trisector of $\angle BAC$ closer to $\overline{AC}$ than $\overline{AB}$ intersects $\overline{BC}$ at $D$. Given that triangle $\vartriangle ABD$ is equilateral with area $1$, compute the area of triangle $\vartriangle ABC$.
[b]p11.[/b] Wanda lists out all the primes less than $100$ for which the last digit of that prime equals the last digit of that prime's square. For instance, $71$ is in Wanda's list because its square, $5041$, also has $1$ as its last digit. What is the product of the last digits of all the primes in Wanda's list?
[b]p12.[/b] How many ways are there to arrange the letters of $SUSBUS$ such that $SUS$ appears as a contiguous substring? For example, $SUSBUS$ and $USSUSB$ are both valid arrangements, but $SUBSSU$ is not.
[b]p13.[/b] Suppose that $x$ and $y$ are integers such that $x \ge 5$, $y \ge 3$, and $\sqrt{x - 5} +\sqrt{y - 3} =
\sqrt{x + y}$. Compute the maximum possible value of $xy$.
[b]p14.[/b] What is the largest integer $k$ divisible by $14$ such that $x^2-100x+k = 0$ has two distinct integer roots?
[b]p15.[/b] What is the sum of the first $16$ positive integers whose digits consist of only $0$s and $1$s?
[b]p16.[/b] Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability $\frac{1}{20}$ while Ajit's coin lands on heads with probability $\frac{1}{22}$ . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does.
[b]p17.[/b] A point is chosen uniformly at random in square $ABCD$. What is the probability that it is closer to one of the $4$ sides than to one of the $2$ diagonals?
[b]p18.[/b] Two integers are coprime if they share no common positive factors other than $1$. For example, $3$ and $5$ are coprime because their only common factor is $1$. Compute the sum of all positive integers that are coprime to $198$ and less than $198$.
[b]p19.[/b] Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates $$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$ Given that Luke's result is greater than $1$, how many possible permutations could he have written?
[b]p20.[/b] Tetrahedron $ABCD$ is drawn such that $DA = DB = DC = 2$, $\angle ADB = \angle ADC = 90^o$, and $\angle BDC = 120^o$. Compute the radius of the sphere that passes through $A$, $B$, $C$, and $D$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
VII Soros Olympiad 2000 - 01, 11.4
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$.
Find the first $200$ decimal digits for the number $a^{2000}$.
2011 Junior Balkan Team Selection Tests - Moldova, 7
In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.
2020 Dutch IMO TST, 2
Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn.
During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses.
Determine which player can always win this game.
2002 USAMTS Problems, 1
The integer $n$, between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is the remainder when $n$ is divided by 4, the digit $d$ is the remainder when $n$ is divied by 5, and the digit $e$ is the reminader when $n$ is divided by 6. Find $n$.
Today's calculation of integrals, 876
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
2018 Mediterranean Mathematics OIympiad, 1
Let $a_1, a_2, ..., a_n$ be more than one real numbers, such that $0\leq a_i\leq \frac{\pi}{2}$. Prove that
$$\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.$$
Kyiv City MO 1984-93 - geometry, 1984.9.5
Using a ruler with a length of $20$ cm and a compass with a maximum deviation of $10$ cm to connect the segment given two points lying at a distance of $1$ m.
1979 IMO Longlists, 40
A polynomial $P(x)$ has degree at most $2k$, where $k = 0, 1,2,\cdots$. Given that for an integer $i$, the inequality $-k \le i \le k$ implies $|P(i)| \le 1$, prove that for all real numbers $x$, with $-k \le x \le k$, the following inequality holds:
\[|P(x)| < (2k + 1)\dbinom{2k}{k}\]
1985 AMC 12/AHSME, 29
In their base $ 10$ representation, the integer $ a$ consists of a sequence of $ 1985$ eights and the integer $ b$ consists of a sequence of $ 1985$ fives. What is the sum of the digits of the base 10 representation of $ 9ab$?
$ \textbf{(A)}\ 15880 \qquad \textbf{(B)}\ 17856 \qquad \textbf{(C)}\ 17865 \qquad \textbf{(D)}\ 17874 \qquad \textbf{(E)}\ 19851$
Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.3
In the quadrilateral $ABCD$ it is known that $\angle ABD= \angle DBC$ and $AD= CD$. Let $DH$ be the altitude of $\vartriangle ABD$. Prove that $| BC - BH | = HA$.
(Hilko Danilo)
2023 Brazil Team Selection Test, 1
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
2017 Simon Marais Mathematical Competition, A1
The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.
2007 Hanoi Open Mathematics Competitions, 8
Let $ABC$ be an equilateral triangle. For a point $M$ inside $\vartriangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.
2001 Spain Mathematical Olympiad, Problem 2
Let $P$ be a point on the interior of triangle $ABC$, such that the triangle $ABP$ satisfies $AP = BP$. On each of the other sides of $ABC$, build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$ and $$CR = RA.$$
Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.
1966 IMO Longlists, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
2021 Iran MO (2nd Round), 2
Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
2010 LMT, 36
Write down one of the following integers: $1, 2, 4, 8, 16.$ If your team is the only one
that submits this integer, you will receive that number of points; otherwise, you receive zero.
[b][color=#f00]There's no real way to solve this but during the competition, each of the 5 available scores were submitted at least twice by the 16 teams competing. [/color][/b]
2018 PUMaC Number Theory A, 8
Let $p$ be a prime. Let $f(x)$ be the number of ordered pairs $(a, b)$ of positive integers less than $p$, such that $a^b \equiv x \pmod p$. Suppose that there do not exist positive integers $x$ and $y$, both less than $p$, such that $f(x) = 2f(y)$, and that the maximum value of $f$ is greater than $2018$. Find the smallest possible value of $p$.
2004 Purple Comet Problems, 23
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?
2019 LIMIT Category C, Problem 4
Which of the following are true?
$\textbf{(A)}~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3$
$\textbf{(B)}~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3$
$\textbf{(C)}~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0$
$\textbf{(D)}~\text{None of the above}$