Found problems: 85335
2018 Purple Comet Problems, 6
Triangle $ABC$ has $AB = AC$. Point $D$ is on side $\overline{BC}$ so that $AD = CD$ and $\angle BAD = 36^o$. Find the degree measure of $\angle BAC$.
2023 China Team Selection Test, P3
(1) Let $a,b$ be coprime positive integers. Prove that there exists constants $\lambda$ and $\beta$ such that for all integers $m$,
$$\left| \sum\limits_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\}\left\{ \frac{bk}{m} \right\} - \lambda m \right| \le \beta$$
(2) Prove that there exists $N$ such that for all $p>N$ (where $p$ is a prime number), and any positive integers $a,b,c$ positive integers satisfying $p\nmid (a+b)(b+c)(c+a)$, there are at least $\lfloor \frac{p}{12} \rfloor$ solutions $k\in \{1,\cdots,p-1\}$ such that $$ \left\{\frac{ak}{p}\right\} + \left\{\frac{bk}{p}\right\} + \left\{\frac{ck}{p}\right\} \le 1 $$
2010 Junior Balkan Team Selection Tests - Romania, 4
The plan considers $51$ points of integer coordinates, so that the distances between any two points are natural numbers. Show that at least $49\%$ of the distances are even.
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]
2007 Today's Calculation Of Integral, 241
1.Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of a parabola $ y \equal{} ax^2 \plus{} bx \plus{} c\ (a\neq 0)$ and the line $ y \equal{} ux \plus{} v$.
Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a|}{6}(\beta \minus{} \alpha )^3}$.
2. Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of parabolas $ y \equal{} ax^2 \plus{} bx \plus{} c$ and $ y \equal{} px^2 \plus{} qx \plus{} r\ (ap\neq 0)$.
Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a \minus{} p|}{6}(\beta \minus{} \alpha )^3}$.
1987 IMO Longlists, 14
Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that
\[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]
Cono Sur Shortlist - geometry, 2012.G2
Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.
2023 AMC 10, 12
When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive?
$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$
2014 PUMaC Algebra B, 1
Evaluate $\tfrac1{\sqrt1+\sqrt2}+\tfrac1{\sqrt2+\sqrt3}+\cdots+\tfrac1{\sqrt{1368}+\sqrt{1369}}$.
Cono Sur Shortlist - geometry, 2021.G4
Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.
2022 Azerbaijan BMO TST, A2
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
2024 Greece National Olympiad, 4
Prove that there exists an integer $n \geq 1$, such that number of all pairs $(a, b)$ of positive integers, satisfying $$\frac{1}{a-b}-\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ exceeds $2024.$
2022 Thailand TST, 3
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
2021 IMO Shortlist, G5
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
2000 Stanford Mathematics Tournament, 9
Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $ x\equal{}4$ and $ y\equal{}8$?
2006 Mediterranean Mathematics Olympiad, 4
Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality
\[ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 \]
1980 AMC 12/AHSME, 5
If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0);
draw(P--C--D^^A--B^^Circle(Q,1));
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, N);
label("$D$", D, S);
label("$P$", P, S);
label("$Q$", Q, SE);
label("$60^\circ$", P+0.0.5*dir(30), dir(30));[/asy]
$ \textbf{(A)} \ \frac{\sqrt{3}}{2} \qquad \textbf{(B)} \ \frac{\sqrt{3}}{3} \qquad \textbf{(C)} \ \frac{\sqrt{2}}{2} \qquad \textbf{(D)} \ \frac12 \qquad \textbf{(E)} \ \frac23 $
1988 Romania Team Selection Test, 7
In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$.
[i]Mircea Becheanu[/i]
2010 Finnish National High School Mathematics Competition, 2
Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.
2017 China Team Selection Test, 3
Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions:
($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$;
($ii$)$2017|x_1+...+x_{100}$;
($iii$)$2017|x_1^2+...+x_{100}^2$.
1978 IMO Longlists, 41
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$
2016 PUMaC Number Theory A, 6
Find the sum of the four smallest prime divisors of $2016^{239} - 1$.
2003 Portugal MO, 2
An architect designed a hexagonal column with $37$ metal tubes of equal thickness. The figure shows the cross-section of this column. Is it possible to build a similar column whose number of tubes ends in $2003$?
[img]https://cdn.artofproblemsolving.com/attachments/6/a/eb5714d2324aac8b78042d1f48f03b74ab0d78.png[/img]
2020 Brazil EGMO TST, 1
Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.
2019 LIMIT Category A, Problem 11
$z$ is a complex number and $|z|=1$ and $z^2\ne1$. Then $\frac z{1-z^2}$ lies on
$\textbf{(A)}~\text{a line not through origin}$
$\textbf{(B)}~\text{|z|=2}$
$\textbf{(C)}~x-\text{axis}$
$\textbf{(D)}~y-\text{axis}$