Found problems: 85335
2021 Girls in Math at Yale, R3
7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers:
[list]
[*] Exactly one of them is a multiple of $2$;
[*] Exactly one of them is a multiple of $3$;
[*] Exactly one of them is a multiple of $5$;
[*] Exactly one of them is a multiple of $7$;
[*] Exactly one of them is a multiple of $11$.
[/list]
What is the maximum possible sum of the integers that Peggy picked?
8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$?
9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.
1987 India National Olympiad, 7
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
2016 ASDAN Math Tournament, 7
Let $x$, $y$, and $z$ be real numbers satisfying the equations
\begin{align*}
4x+2yz-6z+9xz^2&=4\\
xyz&=1.
\end{align*}
Find all possible values of $x+y+z$.
2021 CCA Math Bonanza, I13
Find the sum of the two smallest odd primes $p$ such that for some integers $a$ and $b$, $p$ does not divide $b$, $b$ is even, and $p^2=a^3+b^2$.
[i]2021 CCA Math Bonanza Individual Round #13[/i]
1998 May Olympiad, 2
Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$
2011 Finnish National High School Mathematics Competition, 1
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.
2017 Korea - Final Round, 4
For a positive integer $n \ge 2$, define a sequence $a_1, a_2, \cdots ,a_n$ as the following.
$$ a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$
(a) Show that $a_1, a_2, \cdots a_n$ are all integers.
(b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.
2018 PUMaC Combinatorics A, 8
Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set
$$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$
Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?
Novosibirsk Oral Geo Oly VIII, 2022.2
A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]
2020 China Second Round Olympiad, 2
Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$
2012 Iran Team Selection Test, 1
Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.
[i]Proposed by Hesam Rajabzade[/i]
2021 Dutch IMO TST, 3
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
2006 Princeton University Math Competition, 9
The curve $y=x^4+2x^3-11x^2-13x+35$ has a bitangent (a line tangent to the curve at two points). What is the equation of the bitangent?
1957 Poland - Second Round, 1
Prove that if $ n $ is an integer, then
$$
\frac{n^5}{120} - \frac{n^3}{24} + \frac{n}{30}$$
is also an integer.
2007 Today's Calculation Of Integral, 220
Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.
2022 Brazil EGMO TST, 2
Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?
2019 IFYM, Sozopol, 6
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$
2006 China National Olympiad, 5
Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$, $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$. Prove that
\[ \left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right). \]
2012 Today's Calculation Of Integral, 771
(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.
(2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.
2019 Korea National Olympiad, 1
The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition.
$a_1=1, a_{n+1}=2019a_{n}+1$
Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)
Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$
2018 CCA Math Bonanza, L4.2
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
[i]2018 CCA Math Bonanza Lightning Round #4.2[/i]
2002 AMC 10, 23
Let \[a=\dfrac{1^2}1+\dfrac{2^2}3+\dfrac{3^2}5+\cdots+\dfrac{1001^2}{2001}\] and \[b=\dfrac{1^2}3+\dfrac{2^2}5+\dfrac{3^2}7+\cdots+\dfrac{1001^2}{2003}.\] Find the integer closest to $a-b$.
$\textbf{(A) }500\qquad\textbf{(B) }501\qquad\textbf{(C) }999\qquad\textbf{(D) }1000\qquad\textbf{(E) }1001$
2024 Taiwan TST Round 2, G
Let $ABC$ be a triangle with $O$ as its circumcenter. A circle $\Gamma$ tangents $OB, OC$ at $B, C$, respectively. Let $D$ be a point on $\Gamma$ other than $B$ with $CB=CD$, $E$ be the second intersection of $DO$ and $\Gamma$, and $F$ be the second intersection of $EA$ and $\Gamma$. Let $X$ be a point on the line $AC$ so that $XB\perp BD$. Show that one half of $\angle ADF$ is equal to one of $\angle BDX$ and $\angle BXD$.
[i]Proposed by usjl[/i]
2025 Ukraine National Mathematical Olympiad, 9.6
The sum of $10$ positive integer numbers is equal to $300$. The product of their factorials is a perfect tenth power of some positive integer. Prove that all $10$ numbers are equal to each other.
[i]Proposed by Pavlo Protsenko[/i]
2020 Iran MO (2nd Round), P6
Divide a circle into $2n$ equal sections. We call a circle [i]filled[/i] if it is filled with the numbers $0,1,2,\dots,n-1$. We call a filled circle [i] good[/i] if it has the following properties:
$i$. Each number $0 \leq a \leq n-1$ is used exactly twice
$ii$. For any $a$ we have that there are exactly $a$ sections between the two sections that have the number $a$ in them.
Here is an example of a good filling for $n=5$ (View attachment)
Prove that there doesn’t exist a good filling for $n=1399$