Found problems: 85335
2018 BMT Spring, Tie 3
Find $$\sum^{k=672}_{k=0} { 2018\choose {3k+2}} \,\, (mod \, 3)$$
1999 Romania National Olympiad, 1
Let $P(x) = 2x^3-3x^2+2$, and the sets:
$$A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}$$ Prove that the sets $A \cap B$ and $A\cap C$ have the same number of elements
2022 Azerbaijan IMO TST, 3
Let $ABC$ be a triangle with circumcircle $\omega$ and $D$ be any point on $\omega.$ Suppose that $P$ is the midpoint of chord $AD$ and points $X, Y$ are chosen on lines $AC, AB$ such that reflections of $B, C$ with respect to $AD$ lie on $XP, YP,$ respectively. If the circumcircle of triangle $AXY$ intersects $\omega$ at $I$ for the second time, prove that $\angle PID$ equals the angle formed by lines $AD$ and $BC.$
[i]Proposed by tenplusten.[/i]
2025 District Olympiad, P3
Determine all functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|wf(z)+zf(w)|=2|zw|$$ for all $w,z\in\mathbb{C}$.
2008 ITest, 25
A cube has edges of length $120\text{ cm}$. The cube gets chopped up into some number of smaller cubes, all of equal size, such that each edge of one of the smaller cubes has an integer length. One of those smaller cubes is then chopped up into some number of $\textit{even smaller}$ cubes, all of equal size. If the edge length of one of those $\textit{even smaller}$ cubes is $n\text{ cm}$, where $n$ is an integer, find the number of possible values of $n$.
2008 China Team Selection Test, 6
Find the maximal constant $ M$, such that for arbitrary integer $ n\geq 3,$ there exist two sequences of positive real number $ a_{1},a_{2},\cdots,a_{n},$ and $ b_{1},b_{2},\cdots,b_{n},$ satisfying
(1):$ \sum_{k \equal{} 1}^{n}b_{k} \equal{} 1,2b_{k}\geq b_{k \minus{} 1} \plus{} b_{k \plus{} 1},k \equal{} 2,3,\cdots,n \minus{} 1;$
(2):$ a_{k}^2\leq 1 \plus{} \sum_{i \equal{} 1}^{k}a_{i}b_{i},k \equal{} 1,2,3,\cdots,n, a_{n}\equiv M$.
1998 Iran MO (2nd round), 1
If $a_1<a_2<\cdots<a_n$ be real numbers, prove that:
\[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]
2011 Math Prize For Girls Problems, 9
Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?
2023 Princeton University Math Competition, A1 / B3
Alien Connor starts at $(0,0)$ and walks around on the integer lattice. Specifically, he takes one step of length one in a uniformly random cardinal direction every minute, unless his previous four steps were all in the same directionin which case he randomly picks a new direction to step in. Every time he takes a step, he leaves toxic air on the lattice point he just left, and the toxic cloud remains there for $150$ seconds. After taking $5$ steps total, the probability that he has not encountered his own toxic waste canb be written as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find $a+b.$
2012 Online Math Open Problems, 1
The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
[i]Author: Ray Li[/i]
2024 Argentina Iberoamerican TST, 4
Find all natural numbers $n \geqslant 2$ with the property that there are two permutations $(a_1, a_2,\ldots, a_n) $ and $(b_1, b_2,\ldots, b_n)$ of the numbers $1, 2,\ldots, n$ such that $(a_1 + b_1, a_2 +b_2,\ldots, a_n + b_n)$ are consecutive natural numbers.
2005 Romania National Olympiad, 1
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.
a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1;
b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.
[i]After an idea of Harazi[/i]
1991 Chile National Olympiad, 2
If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.
2020 LMT Fall, 34
Your answer to this problem will be an integer between $0$ and $100$, inclusive. From all the teams who submitted an answer to this problem, let the average answer be $A$. Estimate the value of $\left\lfloor \frac23 A \right\rfloor$. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\max\left(0,\lfloor15-2\cdot\left|A-E\right|\right \rfloor).\]
[i]Proposed by Andrew Zhao[/i]
2013 CHMMC (Fall), 10
Compute the lowest positive integer $k$ such that none of the numbers in the sequence $$\{1, 1 +k, 1 + k + k^2
, 1 + k + k^2 + k^3, ... \}$$ are prime.
2020 IMEO, Problem 3
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all positive real $x, y$ holds
$$xf(x)+yf(y)=(x+y)f\left(\frac{x^2+y^2}{x+y}\right)$$.
[i]Fedir Yudin[/i]
2007 Finnish National High School Mathematics Competition, 3
There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.
2022 Indonesia Regional, 2
(a) Determine a natural number $n$ such that $n(n+2022)+2$ is a perfect square.
[hide=Spoiler]In case you didn't realize, $n=1$ works lol[/hide]
(b) Determine all natural numbers $a$ such that for every natural number $n$, the number $n(n+a)+2$ is never a perfect square.
2009 AMC 12/AHSME, 2
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
2006 IberoAmerican, 1
In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$
The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$
Prove that the triangle $UMN$ is isosceles.
2007 India Regional Mathematical Olympiad, 2
Let $ a, b, c$ be three natural numbers such that $ a < b < c$ and $ gcd (c \minus{} a, c \minus{} b) \equal{} 1$. Suppose there exists an integer $ d$ such that $ a \plus{} d, b \plus{} d, c \plus{} d$ form the sides of a right-angled triangle. Prove that there exist integers, $ l,m$ such that $ c \plus{} d \equal{} l^{2} \plus{} m^{2} .$
[b][Weightage 17/100][/b]
1990 APMO, 2
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
1995 IMC, 11
a) Prove that every function of the form
$$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$
with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)$.
b) Prove that the function
$$F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)$$
has at least $40$ zeroes in the interval $(0,1000)$.
2007 Today's Calculation Of Integral, 243
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
2012-2013 SDML (Middle School), 11
What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below?
[asy]
size(5cm,0);
draw((0,2)--(0,3));
draw((1,1)--(1,3));
draw((2,0)--(2,3));
draw((3,0)--(3,2));
draw((4,0)--(4,1));
draw((2,0)--(4,0));
draw((1,1)--(4,1));
draw((0,2)--(3,2));
draw((0,3)--(2,3));
[/asy]
$\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$