Found problems: 85335
Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .
34. Find the sum of the ages of everyone who wrote a problem for this year’s HMMT November contest. If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - |X - Y|)$
35. Find the total number of occurrences of the digits $0, 1 \ldots , 9$ in the entire guts round (the official copy). If your
answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - \frac{|X-Y|}{2})$
36. Find the number of positive integers less than $1000000$ which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\text{max}(0, 20 - 80|1 -
\frac{X}{Y}|)$ rounded to the nearest integer.
Prove:
From the set $\{1,2,...,n\}$, one can choose a subset with at most $2 \left\lfloor \sqrt n \right\rfloor +1$ elements such that the set of the pairwise differences from this subset is $\{1,2,...,n-1\}$.
($\left\lfloor x \right\rfloor$ means the greatest integer $\leq x$)
In a triangle $ABC$, let $BD$ be its altitude and let $H$ be its orthocenter. The perpendicular bisector of of $HD$ meets $(BCD)$ at $P, Q$. Prove that $\angle APB+\angle AQB=180^{o}$
Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$.
[img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]
Define the set $P=\{a_1, a_2, a_3, \cdots, a_n\}$ and its arithmetic mean $$C_p = \frac{a_1+a_2+\cdots+a_m}m.$$ If we divide $S = \{1, 2, 3, \cdots, n\}$ into two disjoint subsets $A, B$, compute the greatest possible value of $|C_A-C_B|$. For how many $(A, B)$ is equality attained?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 14)[/i]
Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.
A rook stands in a corner of an $n$ by $n$ chess board. For what $n$, moving alternately along horizontals and verticals, can the rook visit all the cells of the board and return to the initial corner after $n^2$ moves? (A cell is visited only if the rook stops on it, those that the rook “flew over” during the move are not counted as visited.)
Proposed by A. Spivak
Let $BD$ and $CE$ be altitudes of an arbitrary scalene triangle $ABC$ with orthocenter $H$ and circumcenter $O$. Let $M$ and $N$ be the midpoints of sides $AB$, respectively $AC$, and $P$ the intersection point of lines $MN$ and $DE$. Prove that lines $AP$ and $OH$ are perpendicular.
Liana Topan
The solution of the equation $ 7^{x\plus{}7}\equal{}8^x$ can be expressed in the form $ x\equal{}\log_b 7^7$. What is $ b$?
$ \textbf{(A)}\ \frac{7}{15} \qquad
\textbf{(B)}\ \frac{7}{8} \qquad
\textbf{(C)}\ \frac{8}{7} \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{15}{7}$
If $\left(0.67\right)^x=0.5$, then find the value of $16\cdot\left(0.67\right)^{3x}$.
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }16\qquad\text{(D) }64\qquad\text{(E) }128$
Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
For a polynomial $f(x)$ with integer coefficients and degree no less than $1$, prove that there are infinitely many primes $p$ which satisfies the following.
There exists an integer $n$ such that $f(n) \not= 0$ and $|f(n)|$ is a multiple of $p$.
For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$?
Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$
Let $m$ and $n$ be positive integers such that
$$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$
Find the maximum possible value of $m^3 + n^3$.
Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.
A triangle $ABC$ with circumcircle $\omega$ satisfies $|AB| > |AC|$. Points $X$ and $Y$ on $\omega$ are different from $A$, such that the line $AX$ passes through the midpoint of $BC$, $AY$ is perpendicular to $BC$, and $XY$ is parallel to $BC$. Find $\angle BAC$.
Suppose $P(x)$ is a cubic polynomial with integer coefficients such $P(\sqrt{5})=5$ and $P(\sqrt[3]{5})=5\sqrt[3]{5}$.
Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.
In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.
[b]8.[/b] The fi gure below shows a broken piece of a circular plate made of glass.
[img]https://cdn.artofproblemsolving.com/attachments/7/3/a49f60d803f802c54e2295932b34579514b4fe.png[/img]
$C$ is the midpoint of $AB$, and $D$ is the midpoint of arc $AB$. Given that $AB = 24$ cm and $CD = 6$ cm, what is the radius of the plate in centimetres? (The fi gure is not drawn to scale.)
Let $ ABC$ be a triangle, and let the angle bisectors of its angles $ CAB$ and $ ABC$ meet the sides $ BC$ and $ CA$ at the points $ D$ and $ F$, respectively. The lines $ AD$ and $ BF$ meet the line through the point $ C$ parallel to $ AB$ at the points $ E$ and $ G$ respectively, and we have $ FG \equal{} DE$. Prove that $ CA \equal{} CB$.
[i]Original formulation:[/i]
Let $ ABC$ be a triangle and $ L$ the line through $ C$ parallel to the side $ AB.$ Let the internal bisector of the angle at $ A$ meet the side $ BC$ at $ D$ and the line $ L$ at $ E$ and let the internal bisector of the angle at $ B$ meet the side $ AC$ at $ F$ and the line $ L$ at $ G.$ If $ GF \equal{} DE,$ prove that $ AC \equal{} BC.$
Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection