Found problems: 85335
2022 Princeton University Math Competition, A6 / B8
Given a positive integer $\ell,$ define the sequence $\{a^{(\ell)}\}_{n=1}^{\infty}$ such that $a_n^{(\ell)}=\lfloor n + \sqrt[\ell]{n}+\tfrac{1}{2}\rfloor$ for all positive integers $n.$ Let $S$ denote the set of positive integers that appear in all three of the sequences $\{a_n^{(2)} \}_{n=1}^{\infty},$ $\{a_n^{(3)} \}_{n=1}^{\infty},$ and $\{a_n^{(4)} \}_{n=1}^{\infty}.$ Find the sum of the elements of $S$ that lie in the interval $[1,100].$
2006 AMC 8, 9
What is the product of $ \dfrac{3}{2}\times \dfrac{4}{3}\times \dfrac{5}{4}\times \cdots \times \dfrac{2006}{2005}$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 1002 \qquad
\textbf{(C)}\ 1003 \qquad
\textbf{(D)}\ 2005 \qquad
\textbf{(E)}\ 2006$
2022 BMT, 8
Given
$$x_1x_2 \cdots x_{2022} = 1,$$
$$(x_1 +1)(x_2 +1)\cdots (x_{2022} +1)=2,$$
$$\text{and so on},$$
$$(x_1 + 2021) (x_2 + 2021) \cdots (x_{2022} + 2021) = 2^{2021},$$
compute
$$(x_1 +2022)(x_2 +2022) \cdots (x_{2022} +2022).$$
2017 ELMO Shortlist, 4
nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square.
[i]Proposed by Mihir Singhal and Michael Kural[/i]
2006 Taiwan TST Round 1, 2
Let $\mathbb{N}$ be the set of all positive integers. The function $f: \mathbb{N} \to \mathbb{N}$ satisfies
$f(1)=3, f(mn)=f(m)f(n)-f(m+n)+2$ for all $m,n \in \mathbb{N}$.
Prove that $f$ does not exist.
Comment: The original problem asked for the value of $f(2006)$, which obviously does not exist when $f$ does not. This was probably a mistake by the Olympiad committee. Hence the modified problem.
1991 IMO Shortlist, 9
In the plane we are given a set $ E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $ E,$ there exist at least 1593 other points of $ E$ to which it is joined by a path. Show that there exist six points of $ E$ every pair of which are joined by a path.
[i]Alternative version:[/i] Is it possible to find a set $ E$ of 1991 points in the plane and paths joining certain pairs of the points in $ E$ such that every point of $ E$ is joined with a path to at least 1592 other points of $ E,$ and in every subset of six points of $ E$ there exist at least two points that are not joined?
2011 Austria Beginners' Competition, 2
Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$
has the real solutions $x_1$ and $x_2$. In addition, the following two conditions apply:
(i) The numbers $x_1$ and $x_2$ differ from each other by exactly $ 1$.
(ii) The numbers $p$ and $q$ differ from each other by exactly $ 1$.
Show that then $p$, $q$, $x_1$ and $x_2$ are integers.
(G. Kirchner, University of Innsbruck)
2017 VJIMC, 4
Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that
\[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]
2023 ISL, G2
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
2022 BAMO, A
If I have 100 cards with all the numbers 1 through 100 on them, how should I put them in order to create the largest possible number?
2008 AMC 12/AHSME, 18
A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid?
$ \textbf{(A)}\ 392 \qquad
\textbf{(B)}\ 196\sqrt{6} \qquad
\textbf{(C)}\ 392\sqrt2 \qquad
\textbf{(D)}\ 392\sqrt3 \qquad
\textbf{(E)}\ 784$
1997 Switzerland Team Selection Test, 4
4. Let $v$ and $w$ be two randomly chosen roots of the equation $z^{1997} -1 = 0$ (all roots are equiprobable). Find the probability that $\sqrt{2+\sqrt{3}}\le |u+w|$
2002 Denmark MO - Mohr Contest, 3
Two positive integers have the sum $2002$. Can $2002$ divide their product?
2019 Purple Comet Problems, 1
Ivan, Stefan, and Katia divided $150$ pieces of candy among themselves so that Stefan and Katia each got twice as many pieces as Ivan received. Find the number of pieces of candy Ivan received.
2023 Stanford Mathematics Tournament, 5
Suppose $\alpha,\beta,\gamma\in\{-2,3\}$ are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note: $\mathbb{R}_{\ge0}$ is the set of nonnegative real numbers). What is the sum of the possible values of $M$?
1997 IMO Shortlist, 14
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
2020 Online Math Open Problems, 20
Reimu invented a new number base system that uses exactly five digits. The number $0$ in the decimal system is represented as $00000$, and whenever a number is incremented, Reimu finds the leftmost digit (of the five digits) that is equal to the ``units" (rightmost) digit, increments this digit, and sets all the digits to its right to 0. (For example, an analogous system that uses three digits would begin with $000$, $100$, $110$, $111$, $200$, $210$, $211$, $220$, $221$, $222$, $300$, $\ldots$.) Compute the decimal representation of the number that Reimu would write as $98765$.
[i]Proposed by Yannick Yao[/i]
2024 Benelux, 1
Let $a_0,a_1,\dots,a_{2024}$ be real numbers such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.
a) Find the minimum possible value of $$a_0a_1+a_1a_2+\dots+a_{2023}a_{2024}$$
b) Does there exist a real number $C$ such that $$a_0a_1-a_1a_2+a_2a_3-a_3a_4+\dots+a_{2022}a_{2023}-a_{2023}a_{2024} \ge C$$ for all real numbers $a_0,a_1,\dots,a_2024$ such that $\left|a_{i+1}-a_i\right| \le 1$ for $i=0,1,\dots,2023$.
1995 IMO, 5
Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.
2018 China Team Selection Test, 6
Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$
2008 Cuba MO, 1
Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$ Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$
2011 Purple Comet Problems, 15
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
2010 Balkan MO Shortlist, G4
Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$.
(a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$.
(b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.
2015 Iran Team Selection Test, 5
Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ :
$P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$
$n>1$
2005 Alexandru Myller, 3
[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $
[b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $