Found problems: 85335
2016 ASDAN Math Tournament, 25
Find the best rational approximation $x$ to $\sqrt[3]{2016}$ such that $|x-\sqrt[3]{2016}|$ is as small as possible. You may either find an $x=\tfrac{a}{b}$ where $a,b$ are coprime integers or find a decimal approximation. Let $C$ be the actual answer and $A$ be the answer you submit. Your score will be given by $\lceil10+\tfrac{16.5}{0.1+e^{30|A-C|}}\rceil$, where $\lceil x\rceil$ denote the smallest integer which is $\geq x$.
2008 China Team Selection Test, 1
Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)
2010 Princeton University Math Competition, 2
Let $f(n)$ be the sum of the digits of $n$. Find $\displaystyle{\sum_{n=1}^{99}f(n)}$.
2005 MOP Homework, 7
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.
2019 Iran MO (3rd Round), 1
Given a cyclic quadrilateral $ABCD$. There is a point $P$ on side $BC$ such that $\angle PAB=\angle PDC=90^\circ$. The medians of vertexes $A$ and $D$ in triangles $PAB$ and $PDC$ meet at $K$ and the bisectors of $\angle PAB$ and $\angle PDC$ meet at $L$. Prove that $KL\perp BC$.
2011 Romania Team Selection Test, 1
Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.
1961 All Russian Mathematical Olympiad, 002
Given a rectangle $A_1A_2A_3A_4$. Four circles with $A_i$ as their centres have their radiuses $r_1, r_2, r_3, r_4$; and $r_1+r_3=r_2+r_4<d$, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle.
Prove that you can inscribe a circle into that quadrangle.
2023 All-Russian Olympiad Regional Round, 9.9
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
2019 Dutch BxMO TST, 4
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
2020 Bulgaria Team Selection Test, 1
In acute triangle $\triangle ABC$, $BC>AC$, $\Gamma$ is its circumcircle, $D$ is a point on segment $AC$ and $E$ is the intersection of the circle with diameter $CD$ and $\Gamma$. $M$ is the midpoint of $AB$ and $CM$ meets $\Gamma$ again at $Q$. The tangents to $\Gamma$ at $A,B$ meet at $P$, and $H$ is the foot of perpendicular from $P$ to $BQ$. $K$ is a point on line $HQ$ such that $Q$ lies between $H$ and $K$. Prove that $\angle HKP=\angle ACE$ if and only if $\frac{KQ}{QH}=\frac{CD}{DA}$.
2010 China Team Selection Test, 3
For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.
1980 IMO, 3
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
2004 Iran MO (3rd Round), 23
$ \mathcal F$ is a family of 3-subsets of set $ X$. Every two distinct elements of $ X$ are exactly in $ k$ elements of $ \mathcal F$. It is known that there is a partition of $ \mathcal F$ to sets $ X_1,X_2$ such that each element of $ \mathcal F$ has non-empty intersection with both $ X_1,X_2$. Prove that $ |X|\leq4$.
2021 CCA Math Bonanza, L5.3
Let $N$ be the number of sequences of words (not necessarily grammatically correct) that have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMATHBONANZA. Here is an example of a possible sequence:
[center]
N, NA, NZA, BNZA, BNAZA, BONAZA, BONANZA, CBONANZA, CABONANZA, CAMBONANZA, CAMABONANZA, CAMAHBONANZA, CCAMAHBONANZA, CCAMATHBONANZA.
[/center]
Estimate $\frac{N}{12!}$. An estimate of $E>0$ earns $\max(0,4-2\max(A/E,E/A))$ points, where $A$ is the actual answer. An estimate of $E=0$ earns $0$ points.
[i]2021 CCA Math Bonanza Lightning Round #5.3[/i]
2024 Turkey MO (2nd Round), 2
Let $\triangle ABC$ be an acute triangle, where $H$ is the orthocenter and $D,E,F$ are the feet of the altitudes from $A,B,C$ respectively. A circle tangent to $(DEF)$ at $D$ intersects the line $EF$ at $P$ and $Q$. Let $R$ and $S$ be the second intersection points of the circumcircle of triangle $\triangle BHC$ with $PH$ and $QH$, respectively. Let $T$ be the point on the line $BC$ such that $AT\perp EF$.
Prove that the points $R,S,D,T$ are concyclic.
2021 AMC 12/AHSME Fall, 2
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$
1950 Miklós Schweitzer, 3
For any system $ x_1,x_2,...,x_n$ of positive real numbers, let
$ f_1(x_1,x_2,...,x_n) \equal{} x_1$,
and
$ f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}}$
for $ \nu \equal{} 2,3,...,n$. Show that for any $ \epsilon > 0$, a positive integer $ n_0 < n_0(\epsilon)$ can be found such that for every $ n > n_0$ there exists a system $ x_1',x_2',...,x_n'$ of positive real numbers
with $ x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1$ and $ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon$ for $ \nu \equal{} 1,2,...,n$ .
2001 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.
OMMC POTM, 2022 2
Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$
Proposed by [b]cj13609517288[/b]
2022 Stanford Mathematics Tournament, 2
The straight line $y=ax+16$ intersects the graph of $y=x^3$ at $2$ distinct points. What is the value of $a$?
2002 National Olympiad First Round, 6
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 4
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 8
$
2010 National Olympiad First Round, 16
$11$ different books are on a $3$-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty?
$ \textbf{(A)}\ 75\cdot 11!
\qquad\textbf{(B)}\ 62\cdot 11!
\qquad\textbf{(C)}\ 68\cdot 12!
\qquad\textbf{(D)}\ 12\cdot 13!
\qquad\textbf{(E)}\ 6 \cdot 13!
$
2023 VIASM Summer Challenge, Problem 3
Given an $8 \times 8$ chess board. Each knight is allowed to move between two squares located at opposite vertices of $2 \times 3$ or $3 \times 2$ rectangles. There are four knights that move on the board, evenly start from the same cell $X$ and return to $X$ and then stop. Assume that every square on the chessboard has at least one of these four roosters moving through.
Prove that there exists a square $Y$ that is different from $X$ such that it is moved over no less than twice by the same knight or by different knights.
1988 AMC 12/AHSME, 27
In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer?
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));
draw(unitcircle);
dot(O);
draw(A--B--C--D--A);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$C$",C,dir(C));
label("$D$",D,dir(D));
label("$O$",O,N);
[/asy]
$ \textbf{(A)}\ AB=3, CD=1\qquad\textbf{(B)}\ AB=5, CD=2\qquad\textbf{(C)}\ AB=7, CD=3\qquad\textbf{(D)}\ AB=9, CD=4\qquad\textbf{(E)}\ AB=11, CD=5 $
1984 Miklós Schweitzer, 3
[b]3.[/b] Let $a$ and $b$ be positive integers such that when dividing them by any prime $p$, the remainder of $a$ is always less than or equal to the remainder of $b$. Prove that $a=b$. ([b]N.16[/b])
[P. Erdos, P. P. Pálify]