Found problems: 85335
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 $1980^n$, $n \ge 1$.
2019 Peru IMO TST, 3
Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.
1988 AIME Problems, 11
Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that
\[ \sum_{k = 1}^n (z_k - w_k) = 0. \]
For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.
1991 Arnold's Trivium, 27
Sketch the images of the solutions of the equation
\[\ddot{x}=F(x)-k\dot{x},\; F=-dU/dx\]
in the $(x,E)$-plane, where $E=\dot{x}^2/2+U(x)$, near non-degenerate critical points of the potential $U$.
VMEO III 2006, 11.2
Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)
1974 AMC 12/AHSME, 8
What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$