Found problems: 85335
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]
$\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2$
2010 Malaysia National Olympiad, 7
Let $ABC$ be a triangle in which $AB=AC$ and let $I$ be its incenter. It is known that $BC=AB+AI$. Let $D$ be a point on line $BA$ extended beyond $A$ such that $AD=AI$. Prove that $DAIC$ is a cyclic quadrilateral.
1969 IMO Shortlist, 37
$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$
2009 VTRMC, Problem 7
Does there exist a twice differentiable function $f:\mathbb R\to\mathbb R$ such that $f'(x)=f(x+1)-f(x)$ for all $x$ and $f''(0)\ne0$? Justify your answer.
VMEO III 2006 Shortlist, A6
The symbol $N_m$ denotes the set of all integers not less than the given integer $m$. Find all functions $f: N_m \to N_m$ such that $f(x^2+f(y))=y^2+f(x)$ for all $x,y \in N_m$.
1968 IMO Shortlist, 13
Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ ($A_iA_k = B_iB_k$), prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented.
2016 USAJMO, 6
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$