Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$? $ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

Is it possible to find a set $A$ of eleven positive integers such that no six elements of $A$ have a sum which is divisible by $6$?

Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.

Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$ [i]Proposed by Alex Li[/i]

Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$

Let $x$ be a positive real number. Find the maximum possible value of \[\frac{x^2+2-\sqrt{x^4+4}}{x}.\]

The sequence \( (a_n)_{n\geq 1} \) of positive integers is such that \( a_1 = 1 \) and \( a_{m+n} \) divides \( a_m + a_n \) for any positive integers \( m \) and \( n \). a) Prove that if the sequence is unbounded, then \( a_n = n \) for all \( n \). b) Does there exist a non-constant bounded sequence with the above properties? (A sequence \( (a_n)_{n\geq 1} \) of positive integers is bounded if there exists a positive integer \( A \) such that \( a_n \leq A \) for all \( n \), and unbounded otherwise.)

Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.

Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle. [i](Karl Czakler)[/i]

Prove that there exists a function $ f $ defined and differentiable in the set of all real numbers, satisfying the conditions $|f'(x) - f'(y)| \leq 4|x-y|$.

For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.

(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

Let $n \geq 2$ be a positive integer and $\{x_i\}_{i=0}^n$ a sequence such that not all of its elements are zero and there is a positive constant $C_n$ for which: (i) $x_1+ \dots +x_n=0$, and (ii) for each $i$ either $x_i\leq x_{i+1}$ or $x_i\leq x_{i+1} + C_n x_{i+2}$ (all indexes are assumed modulo $n$). Prove that a) $C_n\geq 2$, and b) $C_n=2$ if and only $2 \mid n$.

If $p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}$ is a polynomial with real coefficients $a_{i},$ then set $$ \Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}. $$ Let $F(x)=3 x^{2}+7 x+2 .$ Find, with proof, a polynomial $g(x)$ with real coefficients such that (i) $g(0)=1,$ and (ii) $\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)$ for every integer $n \geq 1.$

We call a string of characters [i]neat [/i] when it has an even length and its first half is identical to the other half (eg. [i]abab[/i]). We call a string [i]nice [/i] if it can be split on several neat strings (e.g. [i]abcabcdedef [/i]to [i]abcabc[/i], [i]dede[/i], and [i]ff[/i]). By string [i]reduction[/i] we call an operation in which we wipe two identical adjacent characters from the string (e.g. the string [i]abbac[/i] can be reduced to [i]aac[/i] and further to [i]c[/i]). Prove any string containing each of its characters in even numbers can be obtained by a series of reductions from a suitable nice string. (Martin Melicher)

Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$. Compute the largest possible value for the circumradius of $\mathcal T$.

Bill’s age is one third larger than Tracy’s age. In $30$ years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill?

$n$ pairs are formed from $n$ girls and $n$ boys at random. What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$

Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$. [i](Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)[/i] [i]Proposed by Ivan Chan Kai Chin[/i]

Let $a_1$ be any positive integer. For all $i$, write $5^{2020}$ times $a_i$ in base $10$, replace each digit with its remainder when divided by $2$, read off the result in binary, and call that $a_{i+1}$. Prove that $a_N = a_{N+2^{2020}}$ for all sufficiently large $N$.

Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum $$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$ (with $1 \le r \le s \le 2016$) is a composite number, but: a) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$; b) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $GCD(a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$? $GCD(x, y)$ denotes the greatest common divisor of $x$, $y$. Proposed by Matija Bucić

Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .

Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$

Find the number of lattice points that the line $19x+20y=1909$ passes through in Quadrant I.