A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? [asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9)); [/asy] $\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is $w$, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude $a$. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("$h$",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("$w$",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 2 ([b][u]Question 30[/u][/b]): Assume, instead, that the coefficient of sliding friction between each tire and the ground is different: $\mu_1$ for the front tire and $\mu_2$ for the rear tire. Let $\mu_1 = 2\mu_2$. Assume that both tires are skidding: sliding without rotating. What is the maximum value of $a$ so that both tires remain in contact with the ground? $ \textbf{(A)}\ \frac{wg}{h} $ $ \textbf{(B)}\ \frac{wg}{3h} $ $ \textbf{(C)}\ \frac{2wg}{3h} $ $ \textbf{(D)}\ \frac{hg}{2w}$ $ \textbf{(E)}\ \text{none of the above} $

Real numbers $x$ and $y$ satisfy \begin{align*} x^2 + y^2 &= 2023 \\ (x-2)(y-2) &= 3. \end{align*} Find the largest possible value of $|x-y|$. [i]Proposed by Howard Halim[/i]

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

A number from $1$ to $100$ is intended. In what is the smallest number of questions one can surely guess the intended number, if one is allowed to lie once? (Questions are asked like: “Does the intended number belong to such and such a numerical set?” The only possible answers are “Yes” and “No.”)

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

$a,b$ are two different positive real numbers, then which one is the largest? $$A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.$$ $\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}$Not sure.

Towa has a hand of three different red cards and three different black cards. How many ways can Towa pick a set of three cards from her hand that uses at least one card of each color?

Find the dimension of the solution space of the problem $\partial u/\partial \overline{z} = \delta(z - i)$ for $\text{Im } z \ge 0$, $\text{Im } u(z) = 0$ for $\text{Im } z = 0$, $u\to 0$ as $z\to\infty$.

Let $a_i > 0$ for $i=1,2,\dots,n$ and suppose $a_1 + a_2 + \dots + a_n = 1$. Prove that for any positive integer $k$, \[ \left( a_1^k + \frac{1}{a_1^k} \right) \left( a_2^k + \frac{1}{a_2^k} \right) \dots \left( a_n^k + \frac{1}{a_n^k} \right) \ge \left( n^k + \frac{1}{n^k} \right)^n. \]

Let $R$ and $S$ be the numbers defined by \[R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.\]Prove that $R < \dfrac{1}{15} < S$.

Determine all numbers $ \overline{abcd}$ such that $ \overline{abcd}\equal{}11(a\plus{}b\plus{}c\plus{}d)^2$.

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?