Found problems: 85335
2004 AMC 12/AHSME, 15
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 18 \qquad
\textbf{(C)}\ 27 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 45$
1988 IMO Shortlist, 24
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
2020 China Northern MO, BP1
For all positive real numbers $a,b,c$, prove that
$$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$
2003 AMC 10, 8
The second and fourth terms of a geometric sequence are $ 2$ and $ 6$. Which of the following is a possible first term?
$ \textbf{(A)}\ \minus{}\!\sqrt3 \qquad
\textbf{(B)}\ \minus{}\!\frac{2\sqrt3}{3} \qquad
\textbf{(C)}\ \minus{}\!\frac{\sqrt3}{3} \qquad
\textbf{(D)}\ \sqrt3 \qquad
\textbf{(E)}\ 3$
1998 Estonia National Olympiad, 1
Find the last two digits of $11^{1998}$
2005 Pan African, 1
For any positive real numbers $a,b$ and $c$, prove:
\[ \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c} \]
2005 Regional Competition For Advanced Students, 4
Prove: if an infinte arithmetic sequence ($ a_n\equal{}a_0\plus{}nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n\equal{}b_0q^n$) of real numbers.
2015 Korea National Olympiad, 4
For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$.
(i): $n=a_1+a_2+\cdots +a_l$
(ii): $a_1>a_2>\cdots > a_l > 0$.
(iii): $a_l$ is an odd number.
(iv): There are $k$ odd numbers out of $a_i$.
For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$.
Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.
2024 Pan-African, 1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
PEN K Problems, 4
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
2006 China Second Round Olympiad, 8
Let complex number $z = (a+\cos\theta)+(2a-\sin \theta)i$. Find the range of real number $a$ if $|z|\ge 2$ for any $\theta\in \mathbb{R}$.
2022 Iran MO (3rd Round), 3
We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets.
2021 Kyiv City MO Round 1, 9.4
You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ldots$, and $a_k$ numbers of the $k$-th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$, such that the total number of numbers colored in the same color as $x$ is exactly $x$. What are the possible values of $k$?
[i]Proposed by Arsenii Nikolaiev[/i]
2010 Hanoi Open Mathematics Competitions, 7
Let P be the common point of 3 internal bisectors
of a given ABC: The line passing through P and perpendicular
to CP intersects AC and BC at M and N, respectively. If
AP = 3cm, BP = 4cm, compute the value of $\frac{AM}{BN}$ ?
2025 Bulgarian Winter Tournament, 12.3
Determine all functions $f: \mathbb{Z}_{\geq 2025} \to \mathbb{Z}_{>0}$ such that $mn+1$ divides $f(m)f(n) + 1$ for any integers $m,n \geq 2025$ and there exists a polynomial $P$ with integer coefficients, such that $f(n) \leq P(n)$ for all $n\geq 2025$.
2008 Thailand Mathematical Olympiad, 5
Let $P(x)$ be a polynomial of degree $2008$ with the following property: all roots of $P$ are real, and for all real $a$, if $P(a) = 0$ then $P(a+ 1) = 1$. Prove that P must have a repeated root.
KoMaL A Problems 2023/2024, A. 878
Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear.
Proposed by [i]Vilmos Molnár-Szabó[/i], Budapest
2017 USA TSTST, 4
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$.
[i]Proposed by Mark Sellke[/i]
1996 French Mathematical Olympiad, Problem 5
Let $n$ be a positive integer. We say that a natural number $k$ has the property $C_n$ if there exist $2k$ distinct positive integers $a_1,b_1,\ldots,a_k,b_k$ such that the sums $a_1+b_1,\ldots,a_k+b_k$ are distinct and strictly smaller than $n$.
(a) Prove that if $k$ has the property $C_n$ then $k\le \frac{2n-3}{5}$.
(b) Prove that $5$ has the property $C_{14}$.
(c) If $(2n-3)/5$ is an integer, prove that it has the property $C_n$.
1955 Moscow Mathematical Olympiad, 305
$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?
1989 Irish Math Olympiad, 1
Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.
XMO (China) 2-15 - geometry, 6.5
As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]
2014 Purple Comet Problems, 14
Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.
2014 ASDAN Math Tournament, 7
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.
2018 239 Open Mathematical Olympiad, 8-9.6
Petya wrote down 100 positive integers $n, n+1, \ldots, n+99$, and Vasya wrote down 99 positive integers $m, m-1, \ldots, m-98$. It turned out that for each of Petya's numbers, there is a number from Vasya that divides it. Prove that $m>n^3/10, 000, 000$.
[i]Proposed by Ilya Bogdanov[/i]