Found problems: 85335
2009 India Regional Mathematical Olympiad, 1
Let $ ABC$ be a triangle in which $ AB \equal{} AC$ and let $ I$ be its in-centre. Suppose $ BC \equal{} AB \plus{} AI$. Find $ \angle{BAC}$
2005 India National Olympiad, 4
All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.
2019 Bosnia and Herzegovina Junior BMO TST, 2
$2.$ Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear.
[hide]https://artofproblemsolving.com/community/c6h605458p3596629
https://artofproblemsolving.com/community/c6h1294020p6857833[/hide]
2019 Peru MO (ONEM), 2
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.
1988 IMO Longlists, 19
Let $Z_{m,n}$ be the set of all ordered pairs $(i,j)$ with $i \in {1, \ldots, m}$ and $j \in {1, \ldots, n}.$ Also let $a_{m,n}$ be the number of all those subsets of $Z_{m,n}$ that contain no 2 ordered pairs $(i_1,j_1)$ and $(i_2,j_2)$ with $|i_1 - i_2| + |j_1 - j_2| = 1.$ Then show, for all positive integers $m$ and $k,$ that \[ a^2_{m, 2 \cdot k} \leq a_{m, 2 \cdot k - 1} \cdot a_{m, 2 \cdot k + 1}. \]
2006 Princeton University Math Competition, 1
$A,B,C,D,E$, and $F$ are points of a convex hexagon, and there is a circle such that $A,B,C,D,E$, and $F$ are all on the circle. If $\angle ABC = 72^o$, $\angle BCD = 96^o$, $\angle CDE = 118^o$, and $\angle DEF = 104^o$, what is $\angle EFA$?
2005 China Team Selection Test, 3
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\]
Prove that
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
2023 MIG, 11
A [i]semi-palindrome[/i] is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. What is the difference between the largest semi-palindrome and smallest semi-palindrome?
$\textbf{(A) } 7979\qquad\textbf{(B) } 8080\qquad\textbf{(C) } 8181\qquad\textbf{(D) } 8484\qquad\textbf{(E) } 8989$
2023 CMIMC Algebra/NT, 10
For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$.
[i]Proposed by David Tang[/i]
Oliforum Contest V 2017, 3
Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$
(Paolo Leonetti)
2011 Postal Coaching, 4
Let $a, b, c$ be positive integers for which \[ac = b^2 + b + 1\] Prove that the equation
\[ax^2 - (2b + 1)xy + cy^2 = 1\]
has an integer solution.
1955 Miklós Schweitzer, 7
[b]7.[/b] Prove that for any odd prime number $p$, the polynomial
$2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$
is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b]
*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594
2010 National Olympiad First Round, 11
At most how many points with integer coordinates are there over a circle with center of $(\sqrt{20}, \sqrt{10})$ in the $xy$-plane?
$ \textbf{(A)}\ 8
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None}
$
2024 Mexican Girls' Contest, 5
Consider the acute-angled triangle \(ABC\). The segment \(BC\) measures 40 units. Let \(H\) be the orthocenter of triangle \(ABC\) and \(O\) its circumcenter. Let \(D\) be the foot of the altitude from \(A\) and \(E\) the foot of the altitude from \(B\). Additionally, point \(D\) divides the segment \(BC\) such that \(\frac{BD}{DC} = \frac{3}{5}\). If the perpendicular bisector of segment \(AC\) passes through point \(D\), calculate the area of quadrilateral \(DHEO\).
1997 AMC 8, 25
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$
2024 IFYM, Sozopol, 6
The positive integers \( a \), \( b \), \( c \), \( d \) are such that \( (a+c)(b+d) = (ab-cd)^2 \). Prove that \( 4ad + 1 \) and \( 4bc + 1 \) are perfect squares of natural numbers.
2011 Iran MO (3rd Round), 2
prove that the number of permutations such that the order of each element is a multiple of $d$ is $\frac{n!}{(\frac{n}{d})!d^{\frac{n}{d}}} \prod_{i=0}^{\frac{n}{d}-1} (id+1)$.
[i]proposed by Mohammad Mansouri[/i]
2006 AMC 8, 5
Points $ A, B, C$ and $ D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?
[asy]size(100);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle,linewidth(1));
draw((0,1)--(1,2)--(2,1)--(1,0)--cycle);
label("$A$", (1,2), N);
label("$B$", (2,1), E);
label("$C$", (1,0), S);
label("$D$", (0,1), W);[/asy]
$ \textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 24 \qquad
\textbf{(D)}\ 30 \qquad
\textbf{(E)}\ 40$
2006 IMC, 1
Let $V$ be a convex polygon.
(a) Show that if $V$ has $3k$ vertices, then $V$ can be triangulated such that each vertex is in an odd number of triangles.
(b) Show that if the number of vertices is not divisible with 3, then $V$ can be triangulated such that exactly 2 vertices have an even number of triangles.
2010 Today's Calculation Of Integral, 575
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
2018 Iran Team Selection Test, 2
Determine the least real number $k$ such that the inequality
$$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$
holds for all real numbers $a,b,c$.
[i]Proposed by Mohammad Jafari[/i]
2022 USAMTS Problems, 3
A positive integer $N$ is called [i]googolicious[/i] if there are exactly $10^{100}$ positive integers $x$ that satisfy \[\left\lfloor \frac{N}{\left\lfloor \frac{N}{x} \right\rfloor } \right\rfloor = x,\] where $z$ denotes the greatest integer less than $z.$ Find, with proof, all googolicious integers $N.$
1997 AMC 8, 13
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$\textbf{(A)}\ 31\% \qquad \textbf{(B)}\ 32\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 35\% \qquad \textbf{(E)}\ 95\%$
2006 Princeton University Math Competition, 10
If $a_1, ... ,a_{12}$ are twelve nonzero integers such that $a^6_1+...·+a^6_{12} = 450697$, what is the value of $a^2_1+...+a^2_{12}$?
2013 AMC 8, 3
What is the value of $4 \cdot (-1+2-3+4-5+6-7+\cdots+1000)$?
$\textbf{(A)}\ -10 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 500 \qquad \textbf{(E)}\ 2000$