Found problems: 85335
2015 JBMO Shortlist, NT4
Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\]
Proposed by Moldova
CNCM Online Round 2, 2
There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$?
Proposed by Minseok Eli Park (wolfpack)
2004 Korea Junior Math Olympiad, 2
For $n\geq3$ define $S_n=\{1, 2, ..., n\}$. $A_1, A_{2}, ..., A_{n}$ are given subsets of $S_n$, each having an even number of elements. Prove that there exists a set $\{i_1, i_2, ..., i_t\}$, a nonempty subset of $S_n$ such that
$$A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset$$
(For two sets $A, B$, we define $\Delta$ as $A \Delta B=(A\cup B)-(A\cap B)$)
1969 IMO Shortlist, 13
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
1966 Miklós Schweitzer, 6
A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.)
[i]A.Csaszar[/i]
Today's calculation of integrals, 767
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$
Evaluate $\int_0^1 f(t)dt.$
1990 IMO Longlists, 53
Find the real solution(s) for the system of equations
\[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]
2012 Kazakhstan National Olympiad, 2
Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.
2002 China Western Mathematical Olympiad, 1
Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value.
2009 Cuba MO, 3
In each square of an $n \times n$ board with $n\ge 2$, an integer is written not null. Said board is called [i]Inca [/i] if for each square, the number written on it is equal to the difference of the numbers written on two of its neighboring squares (with a common side). For what values of $n$, can you get [i]Inca[/i] boards ?
2002 ITAMO, 2
The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane.
Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).
2013 Math Prize For Girls Problems, 2
When the binomial coefficient $\binom{125}{64}$ is written out in base 10, how many zeros are at the rightmost end?
1996 North Macedonia National Olympiad, 3
Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$
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2003 IMO Shortlist, 2
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:
(i) move the last digit of $a$ to the first position to obtain the numb er $b$;
(ii) square $b$ to obtain the number $c$;
(iii) move the first digit of $c$ to the end to obtain the number $d$.
(All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)
Find all numbers $a$ for which $d\left( a\right) =a^2$.
[i]Proposed by Zoran Sunic, USA[/i]
2012 China Team Selection Test, 1
In an acute-angled $ABC$, $\angle A>60^{\circ}$, $H$ is its orthocenter. $M,N$ are two points on $AB,AC$ respectively, such that $\angle HMB=\angle HNC=60^{\circ}$. Let $O$ be the circumcenter of triangle $HMN$. $D$ is a point on the same side with $A$ of $BC$ such that $\triangle DBC$ is an equilateral triangle. Prove that $H,O,D$ are collinear.
2020 CCA Math Bonanza, L1.4
Let $ABC$ be a triangle with $AB=3$, $BC=4$, and $CA=5$. Points $A_1$, $B_1$, and $C_1$ are chosen on its incircle. Compute the maximum possible sum of the areas of triangles $A_1BC$, $AB_1C$, and $ABC_1$.
[i]2020 CCA Math Bonanza Lightning Round #1.4[/i]
1980 IMO, 2
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
2002 USAMTS Problems, 4
A transposition of a vector is created by switching exactly two entries of the vector. For example, $(1,5,3,4,2,6,7)$ is a transposition of $(1,2,3,4,5,6,7).$ Find the vector $X$ if $S=(0,0,1,1,0,1,1)$, $T=(0,0,1,1,1,1,0),$ $U=(1,0,1,0,1,1,0),$ and $V=(1,1,0,1,0,1,0)$ are all transpositions of $X$. Describe your method for finding $X.$
1972 IMO Shortlist, 1
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
2018 Iran MO (1st Round), 22
There are eight congruent $1\times 2$ tiles formed of one blue square and one red square. In how many ways can we cover a $4\times 4$ area with these tiles so that each row and each column has two blue squares and two red squares?
1986 IMO, 3
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
2008 Stanford Mathematics Tournament, 3
Give the positive root(s) of $ x^3 \plus{} 2x^2 \minus{} 2x \minus{} 4$.
2013 IMAR Test, 2
For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?
2002 National High School Mathematics League, 9
Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.
2009 Ukraine National Mathematical Olympiad, 4
[b]а)[/b] Prove that for any positive integer $n$ there exist a pair of positive integers $(m, k)$ such that
\[{k + m^k + n^{m^k}} = 2009^n.\]
[b]b)[/b] Prove that there are infinitely many positive integers $n$ for which there is only one such pair.