Someone draws at least three lines on paper. Each cuts the other lines two by two. No three lines pass through one point. He chooses a line and counts the intersection points on either side of the line. The numbers of intersections turn out to be the same. He chooses another line. Now the intersections number on one side appears to be six times as large as that on the other side. What is the minimum number of lines where this is possible? [hide=original wording of second sentence]De lijnen snijden elkaar twee aan twee.[/hide]

Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

The equation $ x^{n+1} +x=0 $ admits $ 0 $ and $ 1 $ as its unique solutions in a ring of order $ n\ge 2. $ Prove that this ring is a skew field.

Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.

Let $S = \{ a_0, a_1, a_2, a_3, \dots \}$ be a set of positive integers with $1 = a_0 < a_1 < a_2 < a_3 < \dots$. For a subset $T$ of $S$, let $\sigma(T)$ be the sum of the elements of $T$. For instance, $\sigma(\{1, 2, 3\}) = 6$. By convention, $\sigma(\emptyset) = 0$, where $\emptyset$ denotes an empty set. Call a number $n$ representable if there exists a subset $T$ of $S$ such that $\sigma(T) = n$. We aim to prove for any set $S$ satisfying $a_{k+1} \le 2a_k$ for every $k \ge 0$, that all non-negative integers are representable. (a) Prove there is a unique value of $a_1$, and find this value. Use this to determine, with proof, all possible sets $\{a_0, a_1, a_2, a_3 \}$. (Hint: there are 7 possible sets.) [Not for credit] I recommend that you show that for all 7 sets in part (a), every integer between $0$ and $a_3 - 1$ is representable. (Note that this does not depend on the values of $a_4, a_5, a_6, \dots$.) (b) Show that if $a_k \le n \le a_{k+1} - 1$, then $0 \le n - a_k \le a_k - 1$. (c) Prove that any non-negative integer is representable.

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

As shown in the following figure, there is a line segment consisting of five line segments $AB, BC, CD, DE, and EA$ and $10$ intersection points of these five line segments. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the following conditions are satisfied. $\bigstar$ The sum of the four numbers written on each line segment $AB, BC, CD, DE, and EA$ is the same.

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square. (Yu. Blinkov)

Points $P$, $Q$, and $M$ lie on a circle $\omega$ such that $M$ is the midpoint of minor arc $PQ$ and $MP=MQ=3$. Point $X$ varies on major arc $PQ$, $MX$ meets segment $PQ$ at $R$, the line through $R$ perpendicular to $MX$ meets minor arc $PQ$ at $S$, $MS$ meets line $PQ$ at $T$. If $TX=5$ when $MS$ is minimized, what is the minimum value of $MS$? [i]2019 CCA Math Bonanza Team Round #9[/i]

Let $ Y_n$ be a binomial random variable with parameters $ n$ and $ p$. Assume that a certain set $ H$ of positive integers has a density and that this density is equal to $ d$. Prove the following statements: (a) $ \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d$ if $ H$ is an arithmetic progression. (b) The previous limit relation is not valid for arbitrary $ H$. (c) If $ H$ is such that $ P(Y_n \in H)$ is convergent, then the limit must be equal to $ d$. [i]L. Posa[/i]

The number halfway between $\dfrac{1}{6}$ and $\dfrac{1}{4}$ is $\text{(A)}\ \dfrac{1}{10} \qquad \text{(B)}\ \dfrac{1}{5} \qquad \text{(C)}\ \dfrac{5}{24} \qquad \text{(D)}\ \dfrac{7}{24} \qquad \text{(E)}\ \dfrac{5}{12}$

Let $ k$ and $ s$ be positive integers. For sets of real numbers $ \{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $ \{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy \[ \sum^s_{i\equal{}1} \alpha^j_i \equal{} \sum^s_{i\equal{}1} \beta^j_i \quad \forall j \equal{} \{1,2 \ldots, k\}\] we write \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}.\] Prove that if \[ \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}\] and $ s \leq k,$ then there exists a permutation $ \pi$ of $ \{1, 2, \ldots , s\}$ such that \[ \beta_i \equal{} \alpha_{\pi(i)} \quad \forall i \equal{} 1,2, \ldots, s.\]

The ratio of the areas of two concentric circles is $ 1: 3$. If the radius of the smaller is $ r$, then the difference between the radii is best approximated by: $ \textbf{(A)}\ 0.41r \qquad \textbf{(B)}\ 0.73 \qquad \textbf{(C)}\ 0.75 \qquad \textbf{(D)}\ 0.73r \qquad \textbf{(E)}\ 0.75r$

Show that if real numbers $x<1<y$ satisfy the inequality $$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.

Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]

Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.

Let $ABCD$ be a cyclic quadrilateral. Let $ P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $ A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $ B$ and $D$. Prove that $PS = PT = PU = PV$.

Let $n$ be a positive integer. Let $s: \mathbb N \to \{1, \ldots, n\}$ be a function such that $n$ divides $m-s(m)$ for all positive integers $m$. Let $a_0, a_1, a_2, \ldots$ be a sequence such that $a_0=0$ and \[a_{k}=a_{k-1}+s(k) \text{ for all }k\ge 1.\] Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$. [i]Proposed by N.V. Tejaswi[/i]

You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that \[(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},\] where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$, where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.

I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that \[ \dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn \] for all $n > 0$.

A point $D$ lies inside a triangle $ABC$ on the bisector of angle $B$. Let $\omega_{1}$ and $\omega_{2}$ be the circles touching $AD$ and $CD$ at $D$ and passing through $B$; $P$ and $Q$ be the common points of $\omega_{1}$ and $\omega_{2}$ with the circumcircle of $ABC$ distinct from $B$. Prove that the circumcircles of the triangles $PQD$ and $ACD$ are tangent. Proposed by: L Shatunov