Found problems: 85335
2006 AMC 10, 25
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2021 AMC 10 Spring, 24
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$?
$\textbf{(A)}\ \frac{8a^2}{(a+1)^2}\qquad\textbf{(B)}\ \frac{4a}{a+1}\qquad\textbf{(C)}\ \frac{8a}{a+1}\qquad\textbf{(D)}\ \frac{8a^2}{a^2+1}\qquad\textbf{(E)}\ \frac{8a}{a^2+1}$
1997 Romania Team Selection Test, 1
We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle.
[i]Mircea Becheanu[/i]
2018 MIG, 16
A triangle with area $60\text{ units}^2$ has vertices with coordinates of $(-15,x)$, $(0,x)$, and $(25,0)$. Find the largest possible value of $x$.
$\textbf{(A) } {-}8\qquad\textbf{(B) } {-}4\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 16$
2023 Princeton University Math Competition, 2
2. Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.
1999 North Macedonia National Olympiad, 5
If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$
2019 LIMIT Category A, Problem 5
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
1970 Miklós Schweitzer, 8
Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2}
\;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\]
[i]P. Turan[/i]
2020 Taiwan TST Round 1, 1
Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$.
(Vietnam)
2013 Saudi Arabia GMO TST, 1
An acute triangle $ABC$ is inscribed in circle $\omega$ centered at $O$. Line $BO$ and side $AC$ meet at $B_1$. Line $CO$ and side $AB$ meet at $C_1$. Line $B_1C_1$ meets circle $\omega$ at $P$ and $Q$. If $AP = AQ$, prove that $AB = AC$.
2014 Serbia JBMO TST, 2
Let $a,b,c,d$ be the natural numbers such that
$2^a+4^b+5^c=2014^d.$
Find all $(a,b,c,d).$
2025 Harvard-MIT Mathematics Tournament, 20
Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$
2016 District Olympiad, 4
[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
[b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
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$.
2018 IMO Shortlist, N2
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
2023 ELMO Shortlist, A6
Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\]
[i]Proposed by Luke Robitaille[/i]
1998 IberoAmerican, 1
There are representants from $n$ different countries sit around a circular table ($n\geq2$), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$, the maximal number of people that can be sit around the table.
2017-2018 SDML (Middle School), 3
Evaluate the following expression: $$0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000$$ The terms with minus signs are exactly the powers of two.
1991 IMO, 2
Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.
2020 AMC 10, 5
What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$
1992 Poland - Second Round, 4
The circles $k_1$, $k_2$, $k_3$ are externally tangent: $k_1$ to $k_2$ at point $A$, $k_2$ to $k_3$ at point $B$, $k_3$ to $k_4$ at point $C$, $k_4$ to $k_1$ at point $D$. The lines $AB$ and $CD$ intersect at the point $S$. A line $ p $ is drawn through point $ S $, tangent to $ k_4 $ at point $ F $. Prove that $ |SE|=|SF| $.
1997 Vietnam National Olympiad, 1
Let $ k \equal{} \sqrt[3]{3}$.
a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k \plus{} k^2) \equal{} 3 \plus{} k$.
b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k \plus{} k^2) \equal{} 3 \plus{} k$
1966 IMO Shortlist, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
2023 Purple Comet Problems, 9
Find the positive integer $n$ such that $$1 + 2 + 3 +...+ n = (n + 1) + (n + 2) +...+ (n + 35).$$
PEN I Problems, 3
Prove that for any positive integer $n$, \[\left\lfloor \frac{n+1}{2}\right\rfloor+\left\lfloor \frac{n+2}{4}\right\rfloor+\left\lfloor \frac{n+4}{8}\right\rfloor+\left\lfloor \frac{n+8}{16}\right\rfloor+\cdots = n.\]