Found problems: 85335
2012 Princeton University Math Competition, B4
For a set $S$ of integers, define $\max (S)$ to be the maximal element of $S$. How many non-empty subsets $S \subseteq \{1, 2, 3, ... , 10\}$ satisfy $\max (S) \le |S| + 2$?
2019 Belarus Team Selection Test, 2.1
Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$.
Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that
$$
p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.
$$
[i](I. Gorodnin)[/i]
2007 AMC 8, 24
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?
$\textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{4}$
Russian TST 2014, P2
The polygon $M{}$ is bicentric. The polygon $P{}$ has vertices at the points of contact of the sides of $M{}$ with the inscribed circle. The polygon $Q{}$ is formed by the external bisectors of the angles of $M{}.$ Prove that $P{}$ and $Q{}$ are homothetic.
2016 Romania Team Selection Tests, 3
Prove that:
[b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and
[b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.
2020 ASDAN Math Tournament, 2
Sam's cup has a $400$ mL mixture of coffee and milk tea. He pours $200$ mL into Ben's empty cup. Ben then adds 100mL of co ee to his cup and stirs well. Finally, Ben pours $200$ mL out of his cup back into Sam's cup. If the mixture in Sam's cup is now $50\%$ milk tea, then how many milliliters of milk tea were in it originally?
2019 India IMO Training Camp, P2
Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.
1961 IMO Shortlist, 3
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
2011 Postal Coaching, 6
Prove that there exist integers $a, b, c$ all greater than $2011$ such that
\[(a+\sqrt{b})^c=\ldots 2010 \cdot 2011\ldots\]
[Decimal point separates an integer ending in $2010$ and a decimal part beginning with $2011$.]
2024 Australian Mathematical Olympiad, P4
Consider a $2024 \times 2024$ grid of unit squares. Two distinct unit squares are adjacent if they share a common side. Each unit square is to be coloured either black or white. Such a colouring is called $\textit{evenish}$ if every unit square in the grid is adjacent to an even number of black unit squares. Determine the number of $\textit{evenish}$ colourings.
2024 District Olympiad, P1
Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$
2021 AMC 12/AHSME Spring, 1
What is the value of $$2^{1+2+3}-(2^1+2^2+2^3)?$$
$\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$
Proposed by [b]djmathman[/b]
2023 Hong Kong Team Selection Test, Problem 4
Let $x$, $y$, $z$ be real numbers such that $x+y+z \ne 0$. Find the minimum value of
$\frac{|x|+|x+4y|+|y+7z|+2|z|}{|x+y+z|}$
1975 AMC 12/AHSME, 23
In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));
label("A", (0,0), S);
label("B", (2,0), S);
label("C", (2,2), N);
label("D", (0,2), N);
label("M", (1,0), S);
label("N", (2,1), E);
label("O", (1.2, .8));
[/asy]
$ \textbf{(A)}\ \frac{5}{6} \qquad\textbf{(B)}\ \frac{3}{4} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(E)}\ \frac{(\sqrt{3}-1)}{2} $
2014 ISI Entrance Examination, 2
Let us consider a triangle $\Delta{PQR}$ in the co-ordinate plane. Show for every function $f: \mathbb{R}^2\to \mathbb{R}\;,f(X)=ax+by+c$ where $X\equiv (x,y) \text{ and } a,b,c\in\mathbb{R}$ and every point $A$ on $\Delta PQR$ or inside the triangle we have the inequality:
\begin{align*} & f(A)\le \text{max}\{f(P),f(Q),f(R)\} \end{align*}
2000 Harvard-MIT Mathematics Tournament, 7
Suppose you are given a fair coin and a sheet of paper with the polynomial $x^m$ written on it. Now for each toss of the coin, if heads show up, you must erase the polynomial $x^r$ (where $r$ is going to change with time - initially it is $m$) written on the paper and replace it with $x^{r-1}$. If tails show up, replace it with $x^{r+1}$. What is the expected value of the polynomial I get after $m$ such tosses? (Note: this is a different concept from the most probable value)
2001 Vietnam Team Selection Test, 2
Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$ the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.
2016 Bulgaria JBMO TST, 4
Given is equilateral triangle $ABC$ with side length $n \geq 3$. It is divided into $n^2$ identical small equilateral triangles with side length $1$. On every vertex of the triangles there is a number. In a move we can choose a rhombus and add or subtract $1$ from all $4$ numbers on the vertices of the rhombus. Let point $D$ have coordinates $(3,2)$ where $3$ is the number of the row and $2$ is the position on it from left to right. On the vertices $A,B,C,D$ there are $1$'s and on the other vertices there are $0$'s. Is it possible, after some operations, all the numbers to become equal?
2012 Today's Calculation Of Integral, 848
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
2004 Estonia Team Selection Test, 2
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
2016 Tournament Of Towns, 4
30 masters and 30 juniors came onto tennis players meeting .Each master played with one master and 15 juniors while each junior played with one junior and 15 masters.Prove that one can find two masters and two juniors such that these masters played with each other ,juniors -with each other ,each of two masters played with at least one of two juniors and each of two juniors played with at least one of two masters.
1995 Singapore Team Selection Test, 1
Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$,
let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that
(i) $g_n(x) > g_n(y)$ if $x > y > 0$.
(ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.
2019 South East Mathematical Olympiad, 5
Let $S=\{1928,1929,1930,\cdots,1949\}.$ We call one of $S$’s subset $M$ is a [i]red[/i] subset, if the sum of any two different elements of $M$ isn’t divided by $4.$ Let $x,y$ be the number of the [i]red[/i] subsets of $S$ with $4$ and $5$ elements,respectively. Determine which of $x,y$ is greater and prove that.
1984 IMO Longlists, 57
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$
1972 IMO, 3
Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.