Found problems: 85335
1977 Bulgaria National Olympiad, Problem 5
Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.
2009 Miklós Schweitzer, 5
Let $ G$ be a finite non-commutative group of order $ t \equal{} 2^nm$, where $ n, m$ are positive and $ m$ is odd. Prove, that if the group contains an element of order $ 2^n$, then
(i) $ G$ is not simple;
(ii) $ G$ contains a normal subgroup of order $ m$.
2010 Contests, 3
Find the number of $4$-digit numbers (in base $10$) having non-zero digits and which are divisible by $4$ but not by $8$.
2011 HMNT, 1
Find the number of positive integers $x$ less than $100$ for which $$3^x + 5^x + 7^x + 11^x + 13^x + 17^x + 19^x$$ is prime.
2019 ASDAN Math Tournament, 10
Regular hexagon $ABCDEF$ has side length $1$. Given that $P$ is a point inside $ABCDEF$, compute the minimum of $AP \sqrt3 + CP + DP + EP\sqrt3$.
2009 Today's Calculation Of Integral, 412
Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$.
(1) Find $ I_0,\ I_{\minus{}1},\ I_2$.
(2) Find $ I_1$.
(3) Express $ I_{n\plus{}2}$ in terms of $ I_n$.
(4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$.
(5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results.
You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.
1992 Tournament Of Towns, (343) 1
Numbers in an $n$ by $n$ table may be changed by adding $1$ to each number on an arbitrary closed non-selfintersecting “rook path” (a broken line with segments parallel to the borders of the table). Originally $1$’s stand on one of the diagonals, and $0S’s in the other cells of the table. Can one get (after several transformations) a table in which all numbers are equal to each other? (A “rook path” contains all cells through which it passes.)
(AA Egorov)
III Soros Olympiad 1996 - 97 (Russia), 10.5
A circle is drawn on a plane, the center of which is not indicated. On this circle, point $A$ is marked and a second circle with center at $A$ is constructed. The second circle has a radius greater than the radius of the first and intersects the first at two points. Construct the center of the first circle using only a compass, drawing no more than five more circles.
2014 Contests, 2
$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.
2011 China National Olympiad, 2
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.
Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$
2024 Dutch IMO TST, 1
On a $2023 \times 2023$ board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a $90^\circ$ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?
2016 Junior Regional Olympiad - FBH, 1
Find unknown digits $a$ and $b$ such that number $\overline{a783b}$ is divisible with $56$
2008 Pre-Preparation Course Examination, 5
A permutation $ \pi$ is selected randomly through all $ n$-permutations.
a) if \[ C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi\] then prove that $ E(C_a(\pi))\equal{}\frac1a$
b) Prove that if $ \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\}$ the probability that $ \pi$ does not have any cycle with lengths $ a_1,\dots,a_k$ is at most $ \frac1{\sum_{i\equal{}1}^ka_i}$
2015 European Mathematical Cup, 1
$A = \{a, b, c\}$ is a set containing three positive integers. Prove that we can find a set $B \subset A$, $B = \{x, y\}$ such that for all odd positive integers $m, n$ we have $$10\mid x^my^n-x^ny^m.$$
[i]Tomi Dimovski[/i]
1964 AMC 12/AHSME, 25
The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely:
$ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $
2004 Harvard-MIT Mathematics Tournament, 6
$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?
1917 Eotvos Mathematical Competition, 3
Let $A$ and $B$ be two points inside a given circle $k$. Prove that there exist (infinitely many) circles through $A$ and $B$ which lie entirely in $k$.
2016 Postal Coaching, 2
Let $\pi (n)$ denote the largest prime divisor of $n$ for any positive integer $n > 1$. Let $q$ be an odd prime. Show that there exists a positive integer $k$ such that $$\pi \left(q^{2^k}-1\right)< \pi\left(q^{2^k}\right)<\pi \left( q^{2^k}+1\right).$$
2005 Estonia Team Selection Test, 2
On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed:
$\bullet$ each Automorian is loved by some Automorian;
$\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$,
$\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$.
Is it correct to claim that every Automorian honours and loves the same Automorian?
Mid-Michigan MO, Grades 7-9, 2023
[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps?
[b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions?
[b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills.
(a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back.
(b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.)
[b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line.
(b) Do the same with $6$ points.
[b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1984 Canada National Olympiad, 1
Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.
2005 Alexandru Myller, 4
Prove that there exists an undirected graph having $ 2004 $ vertices such that for any $ \in\{ 1,2,\ldots ,1002 \} , $ there exists at least two vertices whose orders are $ n. $
2016 PUMaC Geometry A, 4
Let $\vartriangle ABC$ be a triangle with integer side lengths such that $BC = 2016$. Let $G$ be the centroid of $\vartriangle ABC$ and $I$ be the incenter of $\vartriangle ABC$. If the area of $\vartriangle BGC$ equals the area of $\vartriangle BIC$, find the largest possible length of $AB$.
2014 PUMaC Algebra B, 4
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?
2010 Contests, 3
When phenolphythalein is added to an aqueous solution containing one of the following solutes the solution turns pink. Which solute is present?
${ \textbf{(A)}\ \text{NaCl} \qquad\textbf{(B)}\ \text{KC}_2\text{H}_3\text{O}_2 \qquad\textbf{(C)}\ \text{LiBr} \qquad\textbf{(D)}\ \text{NH}_4\text{NO}_3 } $