The sequence $(a_n)_{n>=1}$ satisfies that : $a_1=a_2=1$ $a_n=7a_{n-1}-a_{n-2}$ ($n>=3$) , prove that : for all positive integer n , number $a_n+2+a_{n+1}$ is a perfect square .

For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

The inscribed circle $\omega$ of triangle $ABC$ with center $I$ touches the sides $AB, BC, CA$ at points $C_1, A_1, B_1$. The circle circumsrcibed around $\vartriangle AB_1C_1$ intersects the circumscribed circle of $ABC$ for second time at the point $K$. Let $M$ be the midpoint $BC$, $L$ be the midpoint of $B_1C_1$. The circle circumsrcibed around $\vartriangle KA_1M$ cuts intersects $\omega$ for second time at the point $T$. Prove that the circumscribed circles of triangles $KLT$ and $LIM$ are tangent.

A jeweler can get an alloy that is $40\%$ gold for $200$ dollars per ounce, an alloy that is $60\%$ gold for $300$ dollar per ounce, and an alloy that is $90\%$ gold for $400$ dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is $50\%$ gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.

$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

A non-isosceles triangle $\triangle ABC$ has incenter $I$ and the incircle hits $BC, CA, AB$ at $D, E, F$. Let $EF$ hit the circumcircle of $CEI$ at $P \not= E$. Prove that $\triangle ABC = 2 \triangle ABP$.

At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle. (A. Zaslavsky)

Let $ D,E,F$ be tangency point of incircle of triangle $ ABC$ with sides $ BC,AC,AB$. $ DE$ and $ DF$ intersect the line from $ A$ parallel to $ BC$ at $ K$ and $ L$. Prove that the Euler line of triangle $ DKL$ passes through Feuerbach point of triangle $ ABC$.

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? $ \textbf{(A)}\ 190\qquad\textbf{(B)}\ 192\qquad\textbf{(C)}\ 211\qquad\textbf{(D)}\ 380\qquad\textbf{(E)}\ 382$

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a $2020$-term strictly increasing geometric sequence with an integer common ratio . Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_0$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_0$, what is the closest integer to $\log_{2} d$?

Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$. Proposed by Karl Czakler

Find last three digits of the number $\frac{2019!}{2^{1009}}$ .

The letters $A,B,C$ and $D$ each represent a different digit, so each of the four-digit numbers $ABCD$, $BCDA$, $CDAB$ and $DABC$ satisfy that its least prime divisor is equal to $11$. Determine all possible values of the sum $$ABCD +BCDA+CDAB+DABC$$ and for each possible value of said sum, give an example of a choice of digits $A,B,C$ and $D$ with which to obtain that value and which satisfies the conditions established above.

A bottle contains $5$ gallons of a $10\%$ solution of oil. How many gallons of pure oil must be added to make a $30\%$ oil solution? (round to the nearest hundredth)

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd? $ \textbf{(A)}\ 5743 \qquad\textbf{(B)}\ 5885 \qquad\textbf{(C)}\ 5979 \qquad\textbf{(D)}\ 6001 \qquad\textbf{(E)}\ 6011 $

One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]