Found problems: 85335
2007 Stanford Mathematics Tournament, 8
If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.
2016 CCA Math Bonanza, I7
Simon is playing chess. He wins with probability 1/4, loses with probability 1/4, and draws with probability 1/2. What is the probability that, after Simon has played 5 games, he has won strictly more games than he has lost?
[i]2016 CCA Math Bonanza Individual #7[/i]
2014 Tournament of Towns., 6
A $3\times 3\times 3$ cube is made of $1\times 1\times 1$ cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features:
1) Projection of this solid on each face of the original cube is a $3\times 3$ square,
2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).
KoMaL A Problems 2020/2021, A. 801
For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]
2008 Postal Coaching, 4
Let $n \in N$ and $k$ be such that $1 \le k \le n$. Find the number of ordered $k$-tuples $(a_1, a_2,...,a_k)$ of integers such the $1 \le a_j \le n$, for $1 \le j \le k$ and [u]either [/u] there exist $l,m \in \{1, 2,..., k\}$ such that $l < m$ but $a_l > a_m$ [u]or [/u] there exists $l \in \{1, 2,..., k\}$ such that $a_l - l$ is an odd number.
2008 Sharygin Geometry Olympiad, 5
(Kiev olympiad, 8--9) Reconstruct the square $ ABCD$, given its vertex $ A$ and distances of vertices $ B$ and $ D$ from a fixed point $ O$ in the plane.
2019 Purple Comet Problems, 23
Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$
2011 Sharygin Geometry Olympiad, 2
Let $ABC$ be a triangle with sides $AB = 4$ and $AC = 6$. Point $H$ is the projection of vertex $B$ to the bisector of angle $A$. Find $MH$, where $M$ is the midpoint of $BC$.
2019 Saudi Arabia Pre-TST + Training Tests, 1.1
Suppose that $x, y, z$ are non-zero real numbers such that $$\begin{cases}x = 2 - \dfrac{y}{z} \\ \\ y = 2 -\dfrac{z}{x} \\ \\ z = 2 -\dfrac{x}{y}.\end{cases}$$
Find all possible values of $T = x + y + z$
2019 Jozsef Wildt International Math Competition, W. 3
Compute $$\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx$$
1982 IMO Longlists, 44
Let $A$ and $B$ be positions of two ships $M$ and $N$, respectively, at the moment when $N$ saw $M$ moving with constant speed $v$ following the line $Ax$. In search of help, $N$ moves with speed $kv$ ($k < 1$) along the line $By$ in order to meet $M$ as soon as possible. Denote by $C$ the point of meeting of the two ships, and set
\[AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.\]
Determine the angle $\angle ABC = \beta$ and time $t$ that $N$ needs in order to meet $M$.
2020 AIME Problems, 2
There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1953 AMC 12/AHSME, 7
The fraction $ \frac{\sqrt{a^2\plus{}x^2}\minus{}(x^2\minus{}a^2)/\sqrt{a^2\plus{}x^2}}{a^2\plus{}x^2}$ reduces to:
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{2a^2}{a^2\plus{}x^2} \qquad\textbf{(C)}\ \frac{2x^2}{(a^2\plus{}x^2)^{\frac{3}{2}}} \qquad\textbf{(D)}\ \frac{2a^2}{(a^2\plus{}x^2)^{\frac{3}{2}}} \qquad\textbf{(E)}\ \frac{2x^2}{a^2\plus{}x^2}$
2005 International Zhautykov Olympiad, 1
For the positive real numbers $ a,b,c$ prove that \[ \frac c{a \plus{} 2b} \plus{} \frac d{b \plus{} 2c} \plus{} \frac a{c \plus{} 2d} \plus{} \frac b{d \plus{} 2a} \geq \frac 43.\]
2018 Morocco TST., 5
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
1978 Germany Team Selection Test, 4
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
1991 Arnold's Trivium, 42
Do the medians of a triangle meet in a single point in the Lobachevskii plane? What about the altitudes?
2018 Belarusian National Olympiad, 10.1
The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$.
2023-24 IOQM India, 29
A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.
2016 Purple Comet Problems, 9
Find the value of $x$ such that $2^{x+3} - 2^{x-3} = 2016$.
2005 Hungary-Israel Binational, 1
Does there exist a sequence of $2005$ consecutive positive integers that
contains exactly $25$ prime numbers?
1956 AMC 12/AHSME, 5
A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 12$
2016 Kosovo National Mathematical Olympiad, 1
Find all three digit numbers such that the square of that number is equal to the sum of their digits in power of $5$ .
2019 Singapore Junior Math Olympiad, 1
In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.
2021 Brazil Team Selection Test, 6
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.