Found problems: 85335
Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, the students will spend an additional $5a$ hours per week studying. If the university invests $b$ million dollars in better food, the students will spend an additional $3b$ hours per week studying. Finally, if the $c$ million dollars are invested in parties, students will be more relaxed and spend $11c - c^2$ more hours per week studying. The university wants to invest its $5$ million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study?
Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}
All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?
Nonempty set $A=\{x|2a+1\leq x\leq 3a-5\},B=\{x|3\leq x\leq22\}$, then the range value of $a$ such that $A\subseteq A\cap B$ is
$\text{(A)}\{a|1\leq a \leq9\}\qquad\text{(B)}\{a|6\leq a \leq9\}\qquad\text{(C)}\{a|a \leq9\}\qquad\text{(D)}\varnothing$
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.
Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!
Let $C$ be a cube and let $f: C \longrightarrow C$ be a surjection with
$|PQ| \geq |f(P)f(Q)|$
for all $P,Q \in C$. Prove that $f$ is an isometry.
In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.
Let $G$ be a graph with $400$ vertices. For any edge $AB$ we call [i]a cuttlefish[/i] the set of all edges from $A$ and $B$ (including $AB$). Each edge of the graph is assigned a value of $1$ or $-1$. It is known that the sum of edges at any cuttlefish is greater than or equal to $1$.
Prove that the sum of the numbers at all edges is at least $-10^4$.
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a bijection and let $k$ be a positive integer such that $|f(x+1)-f(x)| \leq k$ for all positive integers $x$. Show that there exists an integer $d$, such that $f(x)=x+d$ for infinitely many positive integers $x$.
$ 101$ positive integers are written on a line. Prove that we can write signs $ \plus{}$, signs $ \times$ and parenthesis between them, without changing the order of the numbers, in such a way that the resulting expression makes sense and the result is divisible by $ 16!$.
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.
The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.
The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
$\textbf{(A) }44\qquad
\textbf{(B) }55\qquad
\textbf{(C) }77\qquad
\textbf{(D) }99\qquad
\textbf{(E) }110$
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation
$$ 1=2mx(2x-1)(2x-2)(2x-3) $$
are real.
Given is a triangle $ABC$ and two points $D \in AC, E \in BD$ such that $\angle DAE=\angle AED=\angle ABC$. Show that $BE=2CD$ iff $\angle ACB=90^{\circ}$.
Find the product of all positive real solutions to the equation $x^{-x} + x^{\frac{1}{x}} = \frac{2021}{2020}.$
[i]Proposed by Gabriel Wu[/i]
$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.
For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6) = 2$ and $b(23)=5$. If $S = \textstyle\sum_{p=1}^{2007}b(p)$, find the remainder when S is divided by 1000.
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.
A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.
Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF \sim \triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.
$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$. Find $p(y^2-1)$.
An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.
Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.
We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.