Found problems: 85335
1993 China Team Selection Test, 1
For all primes $p \geq 3,$ define $F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}$ and $f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}$, where $\{x\} = x - [x],$ find the value of $f(p).$
2014 Singapore Senior Math Olympiad, 29
Find the number of ordered triples of real numbers $(x,y,z)$ that satisfy the following systems of equations:
$x^2=4y-4,y^2=4z-4,z^2=4x-4$
2001 South africa National Olympiad, 1
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)
2010 Today's Calculation Of Integral, 536
Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.
1969 IMO Longlists, 38
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
2010 Vietnam National Olympiad, 5
Let a positive integer $n$.Consider square table $3*3$.One use $n$
colors to color all cell of table such that
each cell is colored by exactly one color.
Two colored table is same if we can receive them from other by a rotation
through center of $3*3$ table
How many way to color this square table satifies above conditions.
2005 Canada National Olympiad, 2
Let $(a,b,c)$ be a Pythagorean triple, i.e. a triplet of positive integers with $ a^2\plus{}b^2\equal{}c^2$.
$a)$ Prove that $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2>8$.
$b)$ Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}\plus{}\frac{c}{b}\right)^2\equal{}n$.
2010 239 Open Mathematical Olympiad, 7
In a convex quadrilateral $ABCD$, We have $\angle{B} = \angle{D} = 120^{\circ}$. Points $A'$, $B'$ and $C'$ are symmetric to $D$ relative to $BC$, $CA$ and $AB$, respectively. Prove that lines $AA'$, $BB'$ and $CC'$ are concurrent.
1966 IMO Longlists, 46
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
1997 Slovenia National Olympiad, Problem 3
Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.
2016 AIME Problems, 9
Triangle $ABC$ has $AB = 40$, $AC = 31$, and $\sin A = \tfrac15$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$.
1997 Israel National Olympiad, 7
A square with side $10^6$, with a corner square with side $10^{-3}$ cut off, is partitioned into $10$ rectangles. Prove that at least one of these rectangles has the ratio of the greater side to the smaller one at least $9$.
1979 Chisinau City MO, 170
The numbers $a_1,a_2,...,a_n$ ( $n\ge 3$) satisfy the relations $$a_1=a_n = 0, a_{k-1}+ a_{k+1}\le 2a_k \,\,\, (k = 2, 3,..., n-1)$$ Prove that the numbers $a_1,a_2,...,a_n$ are non-negative.
1967 AMC 12/AHSME, 23
If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{no finite number}$
2012 Vietnam National Olympiad, 3
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct.
(a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$
(b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.
1999 Polish MO Finals, 1
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
2022-IMOC, C5
Define a "ternary sequence" is a sequence that every number is $0,1$ or $2$. ternary sequence $(x_1,x_2,x_3,\cdots,x_n)$, define its difference to be $$(|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|)$$ A difference will make the length of the sequence decrease by $1$, so we define the "feature value" of a ternary sequence with length $n$ is the number left after $n-1$ differences. How many ternary sequences has length $2023$ and feature value $0$?
[i]Proposed by CSJL[/i]
2008 Gheorghe Vranceanu, 2
Prove that the only morphisms from a finite symmetric group to the multiplicative group of rational numbers are the identity and the signature.
2015 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.
2005 Olympic Revenge, 4
Let A be a symmetric matrix such that the sum of elements of any row is zero.
Show that all elements in the main diagonal of cofator matrix of A are equal.
2011 Tournament of Towns, 5
We will call a positive integer [i]good [/i] if all its digits are nonzero. A good integer will be called [i]special [/i] if it has at least $k$ digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest $k$ such that any good integer can be turned into any other good integer by such moves?
2015 Iran Team Selection Test, 4
$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.
2018 Mathematical Talent Reward Programme, MCQ: P6
In a class among 80 students number of boys is 40 and number of girls is 40. 50 of the students use spectacles. Which of the following is correct?
[list=1]
[*] Only 10 boys use spectacles
[*] Only 20 girls use spectacles
[*] At most 25 boys do not use spectacles
[*] At most 30 girls do not use spectacles
[/list]
1985 IMO Longlists, 40
Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and
\[\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.\]
where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.
2005 Junior Balkan Team Selection Tests - Moldova, 5
Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.