Found problems: 85335
2015 Mathematical Talent Reward Programme, MCQ: P 4
Let $n$ be an odd integer. Placing no more than one $X$ in each cell of a $n \times n$ grid, what is the greatest number of $X$ 's that can be put on the grid without getting $n$ $X$'s together vertically, horizontally or diagonally?
[list=1]
[*] $2{{n}\choose {2}}$
[*] ${{n}\choose {2}}$
[*] $2n $
[*] $2{{n}\choose {2}}-1$
[/list]
2012 District Olympiad, 3
Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that
$$ AB^2=A-B. $$
[b]a)[/b] Show that the matrix $ I_n+B $ is inversable.
[b]b)[/b] Show that $ AB=BA. $
1961 Miklós Schweitzer, 9
[b]9.[/b] Spin a regular coin repeatedly until the heads and tails turned up both reach the number $k$ ($k= 1, 2, \dots $); denote by $v_k$ the number of the necessary throws. Find the distribution of the random variable $v_k$ and the limit-distribution of the random variable $\frac {v_k -2k}{\sqrt {2k}}$ as $k \to \infty$. [b](P. 10)[/b]
1971 Putnam, A2
Determine all polynomials $P(x)$ such that $P(x^2+1)=(P(x))^2+1$ and $P(0)=0.$
Croatia MO (HMO) - geometry, 2011.7
Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.
2014 Argentina Cono Sur TST, 4
Find all pairs of positive prime numbers $(p,q)$ such that
$p^5+p^3+2=q^2-q$
2001 Moldova National Olympiad, Problem 8
If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality
$$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$
1986 National High School Mathematics League, 1
Let $-1<a<0$, $\theta=\arcsin a$. Then the solution set to the inequality $\sin x<a$ is
$\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}$
$\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}$
$\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}$
$\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}$
2018 District Olympiad, 2
Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.
2016 China Girls Math Olympiad, 4
Let $n$ is a positive integers ,$a_1,a_2,\cdots,a_n\in\{0,1,\cdots,n\}$ . For the integer $j$ $(1\le j\le n)$ ,define $b_j$ is the number of elements in the set $\{i|i\in\{1,\cdots,n\},a_i\ge j\}$ .For example ：When $n=3$ ,if $a_1=1,a_2=2,a_3=1$ ,then $b_1=3,b_2=1,b_3=0$ .
$(1)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^2\ge \sum_{i=1}^{n}(i+b_i)^2.$$
$(2)$ Prove that $$\sum_{i=1}^{n}(i+a_i)^k\ge \sum_{i=1}^{n}(i+b_i)^k,$$
for the integer $k\ge 3.$
1967 Miklós Schweitzer, 2
Let $ K$ be a subset of a group $ G$ that is not a union of lift cosets of a proper subgroup. Prove that if $ G$ is a torsion group or if $ K$ is a finite set, then the subset \[ \bigcap _{k \in K} k^{-1}K\] consists of the identity alone.
[i]L. Redei[/i]
1962 Polish MO Finals, 1
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then
$$\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \ldots + \frac{1}{a_{n-1}a_n} = \frac{n-1}{a_1a_n}.$$
2012 Puerto Rico Team Selection Test, 6
The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position?
1972 Polish MO Finals, 3
Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10}
, \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$
1999 German National Olympiad, 1
Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are
a) natural numbers,
b) integers
2018 Macedonia National Olympiad, Problem 2
Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$.
1969 IMO Shortlist, 50
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
1976 AMC 12/AHSME, 23
For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer
$\textbf{(A) }\text{for all }k\text{ and }n\qquad$
$\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$
$\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$
$\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$
2014 Contests, 1
Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
2010 Harvard-MIT Mathematics Tournament, 9
Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.
1991 Chile National Olympiad, 4
Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.
2021 Czech-Polish-Slovak Junior Match, 3
A [i]cross [/i] is the figure composed of $6$ unit squares shown below (and any figure made of it by rotation).
[img]https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png[/img]
Find the greatest number of crosses that can be cut from a $6 \times 11$ divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).
2025 Macedonian TST, Problem 6
Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define
\[
E(V,n)
\;=\;
\bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}.
\]
For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.
2011 India IMO Training Camp, 3
A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be [i]balanced[/i] if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum.
$a)$ Prove that if there exist [i]balanced[/i] sets of sizes $k,j$ then also a [i]balanced[/i] set of size $k+j-1$.
$b)$ Prove that for all [i]odd[/i] $n\geq 7$ there exist a [i]balanced[/i] set of size $n$.
STEMS 2023 Math Cat A, 8
For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?