Found problems: 85335
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Let $M $be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$. Prove that if $AB = AM,$ then a line passing through $M$ perpendicular to $AD$ passes through the midpoint of the arc $BC$.
2012 Romanian Masters In Mathematics, 1
Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)
[i](Poland) Marek Cygan[/i]
2009 JBMO Shortlist, 2
Five players $(A,B,C,D,E)$ take part in a bridge tournament. Every two players must play (as partners) against every other two players. Any two given players can be partners not more than once per a day. What is the least number of days needed for this tournament?
1995 Korea National Olympiad, Problem 2
find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.
2008 SEEMOUS, Problem 4
Let $n$ be a positive integer and $f:[0,1]\to\mathbb R$ be a continuous function such that
$$\int^1_0x^kf(x)dx=1$$for every $k\in\{0,1,\ldots,n-1\}$. Prove that
$$\int^1_0f(x)^2dx\ge n^2.$$
1989 AMC 12/AHSME, 22
A child has a set of $96$ distinct blocks. Each block is one of $2$ materials ([i]plastic, wood[/i]), $3$ sizes ([i]small, medium, large[/i]), $4$ colors ([i]blue, green, red, yellow[/i]), and $4$ shapes ([i]circle, hexagon, square, triangle[/i]). How many blocks in the set are different from the "[i]plastic medium red circle[/i]" in exactly two ways? (The "[i]wood medium red square[/i]" is such a block.)
$ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 39 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 62 $
2007 Estonia Math Open Junior Contests, 10
Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.
2017 All-Russian Olympiad, 1
$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.
2020 CCA Math Bonanza, I7
Define the binary operation $a\Delta b=ab+a-1$. Compute
\[
10 \Delta(10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10))))))))
\]
where $10$ is written $10$ times.
[i]2020 CCA Math Bonanza Individual Round #7[/i]
1983 National High School Mathematics League, 5
$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then
$\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$
2010 Hanoi Open Mathematics Competitions, 5
Each box in a $2x2$ table can be colored black or white. How many different colorings of the table are there?
(A): $4$, (B): $8$, (C): $16$, (D): $32$, (E) None of the above.
2016 Czech And Slovak Olympiad III A, 4
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$.
Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value
2007 ITest, 38
Find the largest positive integer that is equal to the cube of the sum of its digits.
1954 AMC 12/AHSME, 11
A merchant placed on display some dresses, each with a marked price. He then posted a sign “$ \frac{1}{3}$ off on these dresses.” The cost of the dresses was $ \frac{3}{4}$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was:
$ \textbf{(A)}\ \frac{1}{2} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{4} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{4}$
2002 Hong kong National Olympiad, 3
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
2001 AIME Problems, 3
Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0,$ given that there are no multiple roots.
2006 District Olympiad, 2
Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$.
a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$.
b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
2009 Italy TST, 1
Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations?
i)$k$ is a prime number greater than $2$;
ii) $k$ is odd;
iii) $k$ is even.
2017 Thailand TSTST, 4
The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?
2020 China Girls Math Olympiad, 3
There are $3$ classes with $n$ students in each class, and the heights of all $3n$ students are pairwise distinct. Partition the students into groups of $3$ such that in each group, there is one student from each class. In each group, call the tallest student the [i]tall guy[/i]. Suppose that for any partition of the students, there are at least 10 tall guys in each class, prove that the minimum value of $n$ is $40$.
2015 Saint Petersburg Mathematical Olympiad, 2
$a,b>1$ - are naturals, and $a^2+b,a+b^2$ are primes. Prove $(ab+1,a+b)=1$
2018 ISI Entrance Examination, 4
Let $f:(0,\infty)\to\mathbb{R}$ be a continuous function such that for all $x\in(0,\infty)$, $$f(2x)=f(x)$$
Show that the function $g$ defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$ is a constant function.
2021 South East Mathematical Olympiad, 3
Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence.
Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients.
If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.
1963 Swedish Mathematical Competition., 3
What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?
1980 Dutch Mathematical Olympiad, 2
Find the product of all divisors of $1980^n$, $n \ge 1$.