Found problems: 85335
2014 China Second Round Olympiad, 4
Let $x_1,x_2,\dots,x_{2014}$ be integers among which no two are congurent modulo $2014$. Let $y_1,y_2,\dots,y_{2014}$ be integers among which no two are congurent modulo $2014$. Prove that one can rearrange $y_1,y_2,\dots,y_{2014}$ to $z_1,z_2,\dots,z_{2014}$, so that among \[x_1+z_1,x_2+z_2,\dots,x_{2014}+z_{2014}\] no two are congurent modulo $4028$.
2018 China Northern MO, 8
2 players A and B play the following game with A going first: On each player's turn, he puts a number from 1 to 99 among 99 equally spaced points on a circle (numbers cannot be repeated), and the player who completes 3 consecutive numbers that forms an arithmetic sequence around the circle wins the game. Who has the winning strategy? Explain your reasoning.
1990 All Soviet Union Mathematical Olympiad, 523
Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001$.
2017 Moldova Team Selection Test, 9
Let $$P(X)=a_{0}X^{n}+a_{1}X^{n-1}+\cdots+a_{n}$$ be a polynomial with real coefficients such that $a_{0}>0$ and $$a_{n}\geq a_{i}\geq 0,$$ for all $i=0,1,2,\ldots,n-1.$ Prove that if $$P^{2}(X)=b_{0}X^{2n}+b_{1}X^{2n-1}+\cdots+b_{n-1}X^{n+1}+\cdots+b_{2n},$$ then $P^2(1)\geq 2b_{n-1}.$
2023 All-Russian Olympiad Regional Round, 10.8
The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$
2013 Stanford Mathematics Tournament, 1
In triangle $ABC$, $AC=7$. $D$ lies on $AB$ such that $AD=BD=CD=5$. Find $BC$.
2015 China Western Mathematical Olympiad, 3
Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be positive real numbers such that $\sum_{i=1}^nx_i=1$ .Prove that$$\left(\sum_{i=1}^n\frac{1}{1-x_i}\right)\left(\sum_{1\le i<j\le n} x_ix_j\right)\le \frac{n}{2}.$$
1999 Singapore Team Selection Test, 1
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.
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]